The planar end of the Costa surface can be deformed into a catenoidal end, and the same can be done to the symmetrized Costa surfaces. A full account was first given by David Hoffman and Hermann Karcher in 1995.
When deforming the ends further and approaching the limit, one obtains three planes joined by catenoidal necks.
This is the starting point of a regeneration construction of Martin Traizet, that has led to the first complete embedded minimal surface without symmetries.
D. Hoffman, H. Karcher: Complete embedded minimal surfaces of finite total curvature, in Geometry V, Encyclopedia Math. Sci. 90 Springer, (1997) 5-93.