Yes. Expanding a previous comment into an answer: Let $M$ and $N$ be complex manifolds of the same dimension $n >0$ and let $f: M \to N$ be a holomorphic mapping. If $a \in M$ is an isolated point of its fiber $f^{-1}(f(a))$, then $m_af:=\sup\{\#f_\Omega^{-1}(w): w \in \Delta\}$, where $\Omega$ and $\Delta$ are small enough neighborhoods of $a$ and $f(a)$, respectively, is well defined (does not depend on $\Omega$ and $\Delta$) and moreover is finite. Then one can use the Remmert Open Mapping Theorem, which was already addressed on this site: in the analytic category, finite morphisms are open maps? A good reference is Chapter V of Introduction to Complex Analytic Geometry,

Stanislaw Lojasiewicz, Birkh"auser 1991, ISBN 978-3-0348-7617-9-- especially if sheaves are not your cup of tea.

Yes. Expanding a previous comment into an answer: Let $M$ and $N$ be complex manifolds of the same dimension $n >0$ and let $f: M \to N$ be a holomorphic mapping. If $a \in M$ is an isolated point of its fiber $f^{-1}(f(a))$, then $m_af:=\sup\{\#f_\Omega^{-1}(w): w \in \Delta\}$, where $\Omega$ and $\Delta$ are small enough neighborhoods of $a$ and $f(a)$, respectively, is well defined (does not depend on $\Omega$ and $\Delta$) and moreover is finite. Then one can use the Remmert Open Mapping Theorem, which was already addressed on this site: in the analytic category, finite morphisms are open maps? A good reference is Chapter V of Introduction to Complex Analytic Geometry, Stanislaw Lojasiewicz, Birkhäuser 1991, ISBN 978-3-0348-7617-9-- especially if sheaves are not your cup of tea.