Hilbert space and its different sets of orthonormal bases (ONBs) provide an excellent framework for describing complementarity. In spin-1/2 space we chose the $(z^+,z^-)$ basis and make measurements. After the measurement we obtain objects whose properties wrt the different ONBs $(y^+,y^-)$ and $(x^+,x^-)$ and others are maximally undefined. There are many more examples in higher dimensional scenarios, as well, and all of this is supposed to be well understood.
Question: Why do we always end up with the same number of dimensions? Why is not that choosing basis $(z^+, z^+)$ leaves us with some maximally undefined properties for whose understanding we need, say, a 4-dimensional space with a basis, say, $(a,b,c,d)$?
Note 1: I am not asking why all ONBs in a single Hilbert space have the same number of elements. This is a mathematical fact and as such can be demonstrated by a mathematical proof from the relevant definitions and axioms. I am rather interested to learn why nature is such that a complementary aspect to the once chosen always leads to the same Hilbert space with the same number of dimensions.
Note 2: I realize that "Why is nature so-and-so?" are futile. We do not know. However such a question does makes sense in the following regard: Do we have any experiments or "smoking guns" which make it reasonable to expect certain features? For example: $SU(2)$ being a transformation group on qubit/spin-1/2 space and being the universal cover of $SO(3)$ we do have a "smoking gun" for expecting 3-dimensional space.
Note 3: There is a beautiful theorem by Pia Maria Soler that orthomodular lattices with (quite) some additional structure require real, complex or quaternionic Hilbert space to be implemented. Mathematically this is nice. Physically this makes very much sense: We have lots of experiments which point us to orthomodular lattices. In many experiments nature points out that property lattices should not be modeled as distributive lattices but rather as orthomodular. However I still lack the "smoking gun" forcing the same number of dimensions.
I have a mild suspicion that the answer might be connected with degeneracy of eigenvalues (we in fact have situations with different dimension and nature forces us to use, say, 6 dimensions in one scenario, while in another we might only need, say, 5 dimensions, and this is what produces a degenerate eigenspace in the latter situation where we have "too many" degrees of freedom). However as a mathematician I lack practical examples of degeneracies and thus cannot test my suspicion.
Or, is there another aspect I have completely overlooked or misunderstood?