Consider the first-order ODE system for the vector $X = (x,y) \in \mathbb{R}^2$ ,
$$ \frac{d}{dt} X(t) = (r + R_{\pi/2}) \nabla U = r \nabla U + R_{\pi/2} \nabla U $$
where $R_{\pi/2}$ is the $2\times 2$ rotation matrix of $\pi/2$ in the anti-clockwise direction, i.e.
$$ R_{\pi/2}(a,b)=(-b,a) \, ,$$
$r$ is a real number and $U(x,y)$ is a potential in the plane. Hence, $R_{\pi/2}\nabla U$ is just the gradient of $U$ rotated by 90-degrees (since the gradient is orthogonal to the level sets of $U$, if $r=0$ this means that $X$ moves along the level sets of constant $U$).
I wonder if it is possible to see this equation as an ODE for the complex variable $\Phi(t) = x(t)+i y(t)$. If we represent everything in the complex plane, then $R_{\pi/2} \sim i$ because both perform a rotation by 90-degrees, and we should have
$$ \frac{d}{dt} \Phi(t) = (r + i) F(\Phi) $$
where $F=f( x,y)+i g(x,y)$ is an appropriate complex function (say that $\nabla U = (f,g)$). However, it is not clear to me if this can always be safely done, or I am missing some subtleties...
If the answer is "yes, it can be always done", does $F$ have some nice property since it stems from the gradient of a real function? (Feel free to assume all the nicest properties for $U$.)