Update: now crossposted to MathOverflow: https://mathoverflow.net/questions/419705/maps-that-preserve-winding-numbers
I am looking for a characterisation of the continuous maps on some subset of $A\subseteq \mathbb{C}^2$ that preserve the winding numbers of all paths in $A$, i.e. if $\gamma$ is a path whose image lies in $A$ and $x\in A$ is a point not lying on $\gamma$, then $$\text{ind}_{f\circ \gamma}(f(x)) = \text{ind}_\gamma(x)\ .$$
Translations clearly satisfy this. Multiplications with a non-zero complex number do as well. $\mathbb{R}$-linear maps with positive determinant probably as well. $\mathbb{R}$-linear maps with negative determinant on the other hand will flip the sign of the winding number.
Another example is the function $1/z$, which preserves the winding number of any closed curve on any region that does not contain 0.
But is there any class of functions more general than linear maps that have such properties?
It's okay if it only works for closed curves.