One can replace in Costa’s surface the planar end by an Enneper end, keeping the symmetries of the Costa surface and the Enneper surface.

ce 2.0

Of course one loses embeddedness, but gains a parameter. In one limit, the catenoidal necks pinch off and one obtains the Chen-Gackstatter surface, in the other limit, the Enneper end becomes planar, and one has the Costa surface.

I don’t know whether these surfaces have been described in the literature.

Resources

Mathematica notebook and CDF

PoVRay sources