The "standard" definition of a K3 surface is field independent (unless you are a physicist):
$p_g=1, q=0$, and trivial canonical class.
- Mumford and Bombieri showed that you get (just as in the complex case) a 19 dimensional family of K3 surfaces for any degree . (the 19 dimensional thingy is a deformation theory argument which is completely algebraic)algebraic).
- Deligne showed that all the K3 surface is surfaces in finite characteristics are reductions mod p.
What obviously you obviously don't get is the fact that all these spaces sitting 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.