To rephrase the question more precisely than in the title, I'm interested in functions $g:U\to\mathbb C$ for $U$ open in $\mathbb C$ such that for each $z'\in U$ there's a neighbourhood $U'\subseteq U$ of $z'$ and a holomorphic function $f:{U'}^2\to\mathbb C$ such that $g(z)=f(z,\bar z)$ for all $z\in U'$. So for example the absolute value function $\mathbb C - \{0\}\to\mathbb C$ is an example since $|z|=\sqrt{z\bar z}$ even though there's no global holomorphic squareroot function.
I know that the harmonic functions are precisely the ones which are a sum of a holomorphic and antiholomorphic function, but this is more general than that. I can see that such a function has to be infinitely differentiable, but that's pretty much all I can get. Can all infinitely differentiable functions be written in such a way?
These are just the real-analytic functions. Indeed, since $x=\frac{z+\bar{z}}{2}$ and $y=\frac{z-\bar{z}}{2i}$, any real-analytic function on an open subset of $\mathbb{C}$ can be written locally as a power series in $z$ and $\bar{z}$, and conversely any power series in $z$ and $\bar{z}$ can also be viewed as a power series in $x$ and $y$.