I understand (I think) how the Riemann surface of $\sqrt z$ is constructed, or even that of $\log z$. I can visualize them by transforming a plane according to the inverse of the function, i.e. $z^2$ for the former and $\exp z$ for the latter. See, for example https://www.youtube.com/watch?v=ZBvx4iEZOeA.
But then how do I construct the Riemann surface of $z^2$? In that case, the inverse function is multivalued which means that the transformation is not deterministic. So, should I split the points into two and end up representing them as two distinct points on the surface? As in -1 being transformed into both 1 and -1. Or, should I ignore one of the branches?
Sounds like a trivial question but I could not find any examples where the inverse of the function is multivalued.