Motivated by the "The obvious Fact" part of this answer,, we ask the following question:
First we recall a definition, which is used in the above link:
Definition: A non vanishing vector field $V$ on a manifold $M$ is a geodesible vector field if there is a Riemannian metric $g$ on $M$ such that all trajectories of $V$ are (unparametrized) geodesics with respect to $g$.
Assume that $U$ is an open subset of $\mathbb{C}^n$ and $V:U \to \mathbb{C}^n$ is a non vanishing holomorphic function. This holomorphic function $V$ defines a natural vector field on $U\subseteq \mathbb{R}^{2n}$. (Here we identify $\mathbb{C}^{n}$ with $\mathbb{R}^{2n}$). Is $V$ necessarily a geodesible flow?
Remark: The concept "geodesible flow" can be used to count the number of limit cycles of a planar vector field. In this regard please see the following posts:
Flat Riemannian metrics adapted to quadratic vector fields with center
Limit cycles of quadratic systems and closed geodesics(Finitness of $H(2)$)
