In gradient descent you have a function $f:\mathbb{R} \rightarrow \mathbb{R}$. You choose an initial point $x_0$, and set $$x_{i+1} = x_i - \gamma f'(x_i)$$ Repeating this with small enough step size $\gamma$ tends to lead you to a local minimum of $f$.
Suppose you do the same thing with a complex holomorphic function $f:\mathbb{C} \rightarrow \mathbb{C}$. You choose an initial point $z_0$ and set $z_{i+1} = z_i - \gamma f'(z_i)$. Does repeating this with small enough $\gamma$ tend to lead you towards or around any interesting features of $f$?