The symmetrized Costa surface with two catenoidal ends and one planar end was first described by David Hoffman and William Meeks in 1990.
In retrospect, this generalization looks like a small step. As for the Costa surface, there is not really a period problem to solve besides adjusting the López-Ros factor, and the embeddedness proof is not more difficult, either.
However, these were the first embedded minimal surfaces of finite total curvature of arbitrarily large genus, and thus opened a door into completely new terrain.
Moreover, images of the surface with high genus have the appearance of a catenoid intersected by a plane and desingularized by a bent singly periodic Scherk surface. This has triggered the groundbreaking work of Nikos Kapouleas on gluing constructions and led to research that is still ongoing.
D. Hoffman and W. H. Meeks: Embedded minimal surfaces of finite topology. Annals of Mathematics 131 (1990), 1–34.