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

G(z)=\rho\left(z-\frac1{z}\right), \qquad dh = \left(z-\frac1z\right)\, dz \ .

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:

cat1enn1c

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