Suppose $a \in \mathbb C^n$, $U$ is a neighbourhood of $a$, and $f: U \to \mathbb C^n$ is analytic. Let $b = f(a)$ and suppose also that $f^{-1}(b) = \{a\}$. Must the image of $f$ contain a neighbourhood of $b$?

This would be some sort of local version of the open mapping theorem, which in general is not true for several complex variables. If the Jacobian determinant of $f$ is non-zero then the inverse function applies and we are fine. But this condition is not necessary, for example if $f$ is given by $f(z) = z^2$.