mp.mathematical physics - Clifford Algebra in Dirac Equation

I am slightly confused by this question. The fact that one can formulate the Dirac equation in either (3,1) or (1,3) signature, which have non-isomorphic Clifford algebras and hence Clifford modules of different type (real for (1,3) and quaternionic for (3,1) in my naming conventions), does not mean that one is complexifying the Clifford algebra in formulating the Dirac equation.

Unitarity of the time evolution -- a physical requirement independent of choices -- requires that $i D$, where $D$ is the Dirac operator, be hermitian, and in turn this forces a certain hermiticity condition on the "gamma" matrices, in essence choosing a real form of the complex Clifford algebra. It is the spinor representation which can be taken to be complex since after all wave functions live in the tensor product of the Clifford module with a complex Hilbert space. This is not the same thing as complexifying the Clifford algebra, though.

So to summarise, in the Dirac equation the Clifford algebra is real, but the pinor representation can be taken to be complex.