These are complete minimal surfaces of finite total curvature -8π and were completely classified by Francisco López. The most simple case occurs when both ends have parallel normals, here we have a 1-parameter family controlled by the López-Ros parameter which shrinks the catenoidal neck:
The Enneper-Weierstraß data are
If one tilts the limiting normal of the catenoidal end, its flux deforms the Enneper end. Below is an example with perpendicular limiting normals of the ends:
Resources
F. López: The Classification of Complete Minimal Surfaces with Total Curvature Greater than -12π, Trans. Amer. Math. Soc. 334, 49-73, 1992
Mathematica Notebook
PoVRay Sources
Minimal Surfaces 