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I'm trying to find other forms of the Dirac Equation, but that also satisfy the original definition of the equation.

Are you familiar with Hestene's work?

http://geocalc.clas.asu.edu/html/GAinQM.html

I can't make heads or tails of it myself, but it's definitely off the wall. In particular, his geometric interpretation of the unit imaginary is interesting.

To me it seems that the concept of "fitness functions" seems to touch on Genetic Algorithms, and I guess I think less of regression when I hear it.