cv.complex variables - Approximately holomorphic functions

I've got some thoughts, but they should be treated somewhat as speculation. Harald above brought up the notion of quasiconformality. It would not surprise me if the "approximately holomorphic" functions you described were quasiconformal -- the quasiconformality condition is a very soft condition. I would look at the discussions of it in Hubbard's book on Teichmuller theory and Ahlfors's book on quasiconformal mappings and see if you can prove that it holds.

If it does hold, then I'm pretty certain that that the quasiconformality constant would be $1$. If it is, then you are in luck -- a famous theorem of Weyl says that quasiconformal mappings with quasiconformality constant $1$ are actually holomorphic (well, maybe you wouldn't call it luck, as you then wouldn't actually have a generalization).

This sequence of speculations fits into another important intuition about holomorphic functions, namely that they end up being more differentiable than you might guess a priori. For instance, if $f$ has partial derivatives with respect to $x$ and $y$ and the Cauchy-Riemann conditions hold, then $f$ is automatically infinitely differentiable. You don't even have to assume that the partial derivatives of $f$ are continuous or that $f$ is actually differentiable. The theorem of Weyl I mentioned above is also in this vein, as it says that you can even assume that the partial derivatives of $f$ only exist in the weak sense.