Let $f := u + iv : \mathbb{C} \to \mathbb{C}$ be a complex analytic (holomorphic) function. We know that $u, v : \mathbb{R}^2 \to \mathbb{R}$ are individually open maps. I was wondering whether one can conclude the fact that $f$ is an open map from the corresponding properties of $u, v$.
There seem to be many theorems in complex analysis books which are not actually complex analytic in nature (like the maximum principle, mean value property and several others). When I say not "complex analytic in nature", I mean that the proofs do not use the complex structure of $\mathbb{R}^2$. So I am trying to understand whether the open mapping theorem of complex analysis is actually complex analytic in nature. Sorry if the question is too basic.