I understand what pauli matrices $\sigma_x, \sigma_y, \sigma_z$ do mechanically, but the fact they work still seems magical (not in a good way). It seems like a coincidence that we have 3 spatial dimensions and the current formulation fits in neatly with that fact. What if there were 4,5 or more spatial dimensions but still only two possible spin basis states? How can we systematically find pauli matrices in that case?
A general extension to the above question
In general, I am trying to build a geometric understanding of the fundamental concepts in QM and I think I have been slightly successful so far. For example, I understand why the operator $D$ is anti-hermitian and why $iD$ is hermitian geometrically. Is this approach fundamentally flawed because this won't take me too far? Do QM experts solely rely on the algebraic structure to reason through equations/relations and is that only what I should aim for in my course of study? The algebraic approach seems very dry and does not give me the feeling that I actually understand anything, even though I am able to solve the simpler and medium level textbook exercises.
