(All images created with MATLAB® software)
Riemann Surface for (z2-1)1/2. 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 singularities in the surface are at the branch points, namely: at z=1 and at z=-1.
Picture obtained using the script RiemannSur in the Athena 18.04 MATLAB® Toolkit.
Figure 3: Top view of the Riemann Surface for the function f(z)=(z2-1)1/2