Given a domain $\Omega \subset \mathbb{C}^n$, $\Omega^\prime = \Omega \cap \mathbb{R}^n$, and an analytic function $f: \Omega \longrightarrow \Omega$ such that $f_{|\Omega^\prime}$ is a (real) constant, under which assumptions on $\Omega$ and/or $\Omega^\prime$ can one conclude that $f$ is constant on the whole domain $\Omega$?
The case $n = 1$ is quite classical. For $n>1$, assuming $\Omega$ is a pseudoconvex domain does not really help, and I'm not sure of what can be said about real domains of the zeros of an analytical function of several complex variables.