During "Deicide", Aizen made numerous references to the concept of "Transcendence", comparing it to the relationship between it and Humans and Shinigami as akin to the relationship between a two-dimensional object and three-dimensional objects - because he was in a 'higher plane', Humans and Shinigami could not feel his spiritual power. I think that Aizen was not wrong when he was making this analogy, as he was quite specific when used only the second and third dimensions of objects while ignoring the first dimension.
In Bleach, I believe that the dimensions Aizen made reference to is linked to the nature of the soul itself - in other words, the different "planes of souls". So if Humans and Shinigami occupy the 2nd dimension, while Aizen occupies the 3rd dimension (and Ichigo was able to ascend into a 4th dimension), what would the 1st dimension be?
Looking back with what I know now, I think that answer is clear: the "soul" in "matter" are the "first plane of souls"; living beings and spirits make up the "second plane of souls"; a Transcendental Being like what Aizen became resides in the "third plane of souls", and Ichigo managed to transcend into the "fourth plane of souls" when he "became one with his power".
During "Deicide", Aizen made a curious comment that stuck in my mind and blossomed into thought with the Lost Agent Arc:
"Living beings are strange creatures. They are designed to realize whatever their under-sized hearts desire."
The above is the Viz translation, and translations at Mangahelper give it more depth.
Cnet128 has Aizen say:
when it comes to achieving those petty desires that their crude hearts enshrine... // ...they tend to have remarkable levels of success.
While molokidan translated this:
Despite the nature of what they wish for with their stunted hearts,
5: they are built so that they are able to realize such.
All three point toward one common idea: the Human heart is designed to realize whatever it is the individual desires most. Looking back on this after the Lost Agent arc, I realized Aizen may have been referring to the nature of Fullbring.
Fullbring is a power that allows Humans to manipulate the "souls" in "matter", and each Fullbringer has a unique ability that is closely aligned with a specific desire. By drawing out the souls in matter and boosting it with their own to unleash a greater power, a Fullbringer essentially manipulates "Reason" to achieve their desires.
This "Power of Reason" is a trait possessed by "one-dimensional souls", as it is what defines the Laws of Nature. A Fullbringer can change the "nature of matter" through manipulating the "souls" contained within "matter". To be more accurate, what would define a "two dimensional soul" is the ability to draw out its own power through various methods, something a "one dimensional soul" cannot do.
A "three dimensional soul", on the other hand, defies "Reason" itself.
We saw this occur when Aizen destroyed the Koutotsu. Gin described the Koutotsu as a "Being of Reason/Force of Nature", depending on the translation, and he and Isshin stated that no Shinigami should be able to do anything to it with reiatsu. However, Aizen destroyed the Koutotsu by letting it hit him. The Koutotsu, as a "Being of Reason", was something no being possessing reiatsu should be able to effect, as it was essentially a living embodiment of the Laws of Nature that define the world of Bleach. To even consider destroying it would require one to possess power that stands above and defies "Reason" itself - an action that gives scope to what a Transcendental Being is.
"Transcending one's limits" have been mentioned before in regards to Hollowfication and Hollows becoming Arrancar - even Aizen said Inoue's powers were beyond Human limits - yet neither Visoreds, Arrancar or Fullbringers can possibly be grouped into the same category as a Transcendental Being.
It is at this point where the opposition of souls Urahara brought up applies. While it is Humans and Hollow who are opposites of one another, the process of Hollowfication involves breaking down the barrier between Shinigami and Hollow - otherwise incompatible beings - in order to allow a Shinigami to go beyond their normal limits. However, there exists a natural instability in such a process, leading to "soul suicide" in Shinigami. In regards to Quincy, the opposites of Shinigami, they cannot even gain Hollow power and are out-right destroyed by a Hollow's essence.
As Urahara also explained, the way to stop "soul suicide" in a Visored - and halt Hollowfication in a Quincy altogether - is to introduce an element that corrects the 'imbalance' of a soul caused by such a process. For Visoreds, it is a vaccine of Quincy arrows of light and Human souls; in Masaki's case, Isshin had to become Shinigami-Human and have his soul tied to hers and White's to balance out her Quincy-Hollow.
In contrast, Fullbringers and Arrancar are completely stable, and the reason for this is likely due to the fact Human and Hollow already balance with one another. Fullbring is a Human power derived from Hollow reiryoku; Arrancars are created when a Hollow breaks their mask and regains their Human form. We have yet to see a straight-forward Shinigami-Quincy, but they would likely balance out in a similar fashion.
As a result, a Transcendental Being - or "three dimensional soul" - is likely an entity where the above four soul types are perfectly balanced with one another.
Essentially, it means that a "three dimensional soul" is created when all four types of "two dimensional souls" exist and balance with one another within a single individual. We saw this in "The Lust" with Zangetsu, "Deicide" with Aizen, and I believe we also saw it with Ginjo in "The Lost Agent arc".
However, I once believed that it was simply Shinigami-Hollow-Human balanced in one being.
In support of the Shinigami-Hollow-Human I mentioned earlier, Ginjo told Ichigo that his Shinigami power would fuse with his Fullbring, and that Ichigo would "gain a power transcending the Shinigami". The last time a character talked about "power transcending the Shinigami", he previously destroyed the Koutotsu. Supporting the idea that Ginjo was talking about full-blown Transcendence, when Ichigo released his Bankai after gaining his powers back he shattered Yukio's "chatroom".
An entire team of Captain-level Shinigami were implied to not be able to break into it if they combined all their power, but Ichigo shattered it by releasing his Bankai. It is likely not because Ichigo was stronger than all five of said Captains combined, but because the "nature" of his power allowed him to transcend and defy the laws governing Yukio's "Invaders Must Die", allowing him to interact with it in a way normal Shinigami couldn't.
Also supporting the idea of Shinigami-Human-Hollow Transcendence is Zangetsu during "The Lust", which also indicates a concept of "dominance" in regards to Transcendental Beings. When Zangetsu came out on the Dome, he was more Hollow than Shinigami ("Quincy" Zangetsu was suppressing the Shinigami power), so what we saw behaved and fought like a Hollow. It was likely this entity's power that Aizen used as the basis for his Transcendence, leading to his more Hollow-like appearance with each transformation.
We saw this again in "The Lost Agent arc" between Ichigo and Ginjo. While Ginjo was a Fullbringer who had Shinigami power fuse with his Fullbring i.e. the one he stole from Ichigo, Ichigo was a Shinigami who had his Fullbring fuse with his Shinigami power. In short, Ichigo expressed his power through the medium of a Shinigami, while Ginjo did so through Fullbring, leading to their strikingly different appearances when they entered Bankai - Ichigo remained the same, but Ginjo took on a Hollow-like appearance.
However, Urahara's explanation indicates that a perfect fusion of Shinigami and Hollow balanced by a Human soul isn't enough - there needs to be a Quincy component. "Quincy" Zangetsu provided this for Ichigo, and the idea gives interesting implications for Ginjo - something I'll get into later because of how late it is where I am.
So now we finally move onto the "fourth plane of power/souls" that Ichigo achieved in Deicide. What this "fourth plane of souls" is likely pertains to what the essence behind the Final Getsuga Tenshou was - "becoming Getsuga" i.e. becoming one with his power.
Please note that I do not think this is something exclusive to Transcendents, as Isshin is evidence that Shinigami can "become one with their power". Rather, it indicates that "becoming one with your power" is a natural form of Transcendence accessible to Shinigami. As a result, similar forms likely exist with Hollow, Quincy, and potentially Fullbringer.
Obviously, we have seen what this entails for Shinigami, so I will elaborate on Hollow and Quincy.
In regards to Hollows, I believe this is a state they can only achieve once they become Arrancar, and that it is the Segunda Etapa that Ulquiorra used during "The Lust". The basis of a Resurreccion is to return the Arrancar to its 'true form', usually something representing what they originally looked like as Hollows to varying degrees. But we know there is a Second Stage because of Ulquiorra, and one detail sticks out in my mind regarding his Segunda Etapa - the missing Mask Fragment.
A Mask Fragment often changes with the release of the Resurreccion, but Ulquiorra's Segunda Etapa caused the Mask Fragment to disappear completely. As implied by Neliel's turning into a child after her Mask fragment was broken, there is considerable power stored in the Mask Fragment that most Arrancar do not tap into. I believe that tapping into that power involves completely fusing the power of the Hollow with the Arrancar's Human body, their form of "becoming one with their power".
This is possibly evidenced by Uryu noting how the 'nature' of Ulquiorra's reiatsu was different after he went into Segunda Etapa.
In the case of the Quincy, the answer appears to be getting complicated with the depictions of the Stern Ritter and their power, but I do believe the Letzt Stil is the Quincy's key to their "Transcendence" . Much like the Final Getsuga Tenshou, the Letzt Stil increases the Quincy's power to new heights, but the end result with both of them is that the individual using it loses their powers. A Quincy loses their power as a Quincy but still retains their spiritual power, while a Shinigami loses their spiritual power entirely.
The Letzt Stil centers around increasing the Quincy's ability to collect reishi to such an extent that they exert "dominance over reishi", and an example of the power achievable through this method is seen with Uryu when he used it against Mayuri. I cannot say with certainty that a Quincy "becomes one with their power" because of this, but reishi dominance is a power that goes beyond the normal limits of Quincy, as Mayuri pointed out. But perhaps it is that reishi dominance - considering manipulating reishi is the center of a Quincy's power - that symbolizes them "becoming one with their power".
At this point, a similar power existing for Fullbringers is only speculation, but it would likely entail something related to their ability to manipulate the "souls" in "matter" - maybe even along the lines of Shinigami and Hollow, considering the manner a Fullbringer manifests their power is similar to a Zanpakuto. Rather than channelling their power through an item they have a close attachment to, the height of a Fullbringer's power would involve a manipulation of all the "souls" in the "matter" comprising their body, and channeling their power via that method.
For Shinigami, Hollow, Human and Quincy, this would potentially give them access to the 'third plane of power', but I cannot say that with certainty. For a Transcendental Being, this allows them to enter into the 'fourth plane of power'.