We know some things for real valued functions. Let $X$ be a topological space, and call $Z \subset X$ a zero set if there exists a continuous function $f:X \rightarrow \mathbb{R}$ s.t. $f^{-1}(0) = Z$.
We know:
Every closed is a zero set iff $X$ is perfectly normal
Every closed is an intersection of zero sets iff $X$ is Tychonoff
If $X$ is compact Hausdorff, then every singleton is a zero-set iff $X$ is first countable.
So my question is: Do the same results hold true if we consider zero sets of complex valued functions $f:X \rightarrow \mathbb{C}$? Or what does change?
The motivation is, that for a compact Hausdorff space $X$, I would like to understand which closed sets are zero sets of complex continuous functions $f:X \rightarrow \mathbb{C}$. Or under what hypotheses every closed (/singleton) is such a zero set.