Is there an analogue for bisection (or the golden section method), i.e., solving $f(x) = 0$ when $f:\mathbb{R} \rightarrow \mathbb{C}$.
When $f$ is real-valued, we know (at least one) root exists if there exist a bracket $x_l < x_u$ such that $f(x_l)f(x_u) < 0$, in which case $x_l$ and $x_u$ "sandwich" the root. So we iteratively shrink the bracket in a way that preserves $f(x_l)f(x_u) < 0$.
My question is, is there an analogue for this when $f$ is complex valued. Specifically, is there a condition similar to $f(x_l)f(x_u) < 0$ for "sandwiching" the root?