The torus with two Ennneper ends exists for all higher dihedral symmetry groups:
Above you can see a slightly hard to parse image for n=3, and below two examples for n=5:
For fixed n, the surfaces come in a 1-parameter family. Conjecturally, one limit could be two planes joined by catenoidal necks, and the other could maybe an n-Noid.
All this raises the question: What kind of topology can one have between two Enneper ends? How flexible is one in placing the catenoidal necks? This is somewhat analogous to the old (and still unsolved) problem to determine what topology is possible for a minimal surface between two convex curves in parallel planes.
Minimal Surfaces 