In the same spirit as Michael's answer: Over the complex variables, there is a theorem due to W.F. Osgood (published in his "Lehrbuch der Funktionentheorie", 2. Aufl., Bd. 2, Parte 1, Leipzig 1929) about solvability of $w=f(z)$ for a system of holomorphic functions on a neighborhood of a point $a \in \mathbb{C}^n$, which is an isolated point of the set $\{z:f(z)=b:=f(a)\}$. This is nicely discussed in B.V. Shabat's book "Complex analysis" (part II, section 14, item 44-although I am not sure if this is included in the English translation of the book)-in terms of resultants and Weierstrass' Preparation Theorem. Shabat also refers to: M. Herve, "Several Complex Variables. Local Theory", Oxford 1963.
And there is some information (on the topic of failure of the implicit function theorem) in the book by Aizenberg and Yuzhakov on residue theorems.