Complex geometry and Dirac equation

Complex geometry represents a fundamental ingredient in the formulation of the Dirac equation by the Clifford algebra. The choice of appropriate compl ex geometries is strictly related to the geometric interpretation of the co mplex imaginary unit i = root-1. We discuss two possibilities which appear in the multivector algebra approach: the sigma(123) and sigma(21) complex g eometries. Our formalism provides a set of rules which allows an immediate translation between the complex standard Dirac theory and its version withi n geometric algebra. The problem concerning a double geometric interpretati on for the complex imaginary unit i = root-1 is also discussed.