(All images created with MATLAB® software
Riemann Surface for the square root. This surface is an object in four dimensions and cannot be drawn properly in three dimensions. The self intersections that the picture shows are a by-product of projecting the surface on three dimensions. The only singularity in the surface arises at the origin (the branch point).
Picture obtained using the script RiemannSur in the Athena 18.04 MATLAB® Toolkit.
Figure 1: One view of a square root Riemann Surface