I am wondering if there is any mathematical (or physical, besides the fact that classical quantum mechanics uses complex numbers) justification for why the complexified (1,3) Clifford algebra is used in Dirac's equation. A (the?) key point of special relativity is that spacetime is a real 4-d vector space with an inner product of signature (1,3). But by complexifying the signature becomes irrelevant-- all complex Clifford algebras in a given dimension are isomorphic where for real Clifford algebras, even signatures (p,q) and (q,p) are not isomorphic in general (as a side question: can there be a physical significance to this fact? or do only the spin group and the even subalgebra of the Clifford algebra, which are the same for (p,q) and (q,p), matter? I only hear about spinor bundles and spin structures, never pinor bundles or pin structures).
Thanks and I hope this isn't too physicsy of a question!