Help a lapsed mathematician out here... the complex numbers can be represented as skew-symmetric 2x2 real matrices, right? And this is an isomorphism? So if the 2x2 real skew-symmetric matrices are "the same" as the complex numbers, how do we end up having this issue that comes up in quantum physics where you can only use the actual complex numbers and not the representation?
Any explanation I can give in a comment here will be pretty flawed, if you want more detail than my previous comment I think your best bet is to read the papers at the arxiv links I posted above.
