what the hell are complex numbers doing in the Dirac equation?

As I've pointed out to you before, if you have a problem with physical applications of complex numbers, you should be equally offended by physical applications of matrices, because matrices of the form [[a,-b],[b,a]] are isomorphic to complex numbers. In fact, your problem isn't just with quantum mechanics; if you can't stand complex numbers, you should also have a problem with (for just one example) simple harmonic motion.

In detail: we model a mass attached to a spring with the equation F=-kx: the force F on the mass is proportional to a constant -k times the displacement from the equilibrium position x. But because force is mass times acceleration, and acceleration is the second time derivative of position, this is actually the differential equation x''(t) + (k/m)x(t) = 0, which has the solution x(t) = ae^(i*sqrt(k/m)t) + be^(-i*sqrt(k/m)t) where a and b are arbitrary constants.

It's true that people tend to write this as ccos(sqrt(k/m)t)+dsin(sqrt(k/m)t), but the fact that we use a notation that makes the complex numbers less visible, doesn't change the underlying math. Trig functions are sums of complex exponentials.

Complex numbers are perfectly well-behaved, non-mysterious mathematical entities (consider also MagnetoHydroDynamics's point about algebraic closure); why shouldn't they appear in the Dirac equation?