How to classify K3 surfaces over an arbitrary field k?
The "standard" definition of a K3 surface is field independent (unless you are a physicist):
$p_g=1, q=0$, and trivial canonical class.
What you obviously don't get is the fact that all these spaces sit together in a nice 20 dimensional complex ball. I also don't know if you can carry over any of the recent Kodaira dimension computation of these moduli (which are very analytic in nature).
Reference: Complex algebraic surfaces (Beauville): Chapter VIII and Appendix A.