In his dissertation from 1993, Meinhard Wohlgemuth constructs the a 4-ended embedded minimal surface of genus 2, which he calls CSSCFF. This was the first complete embedded minimal surface with four ends, generalizing Costa’s surface.
It has two catenoidal and two planar ends and the coordinate planes as symmetry planes. The existence proof is involved, it requires to solve a 2-dimensional period problem. An embeddedness proof is still lacking.
Dihedralizations also exist, with the same type of ends but higher genus. The original surface for genus 2 is a borderline case of the Hoffman-Meeks conjecture: A complete, embedded minimal surface of finite total curvature with n ends must have genus at least n-2.
It is possible to deform the inner planar ends into catenoidal ends, that will then eventual intersect the outer catenoidal ends, in contrast to the Costa-Hoffman-Meeks family, which is embedded throughout.
M. Wohlgemuth: Vollständige Minimalflächen höheren Geschlechts und endlicher Totalkrümmung, Dissertation, Bonn, 1993.
M. Wohlgemuth: Minimal Surfaces of Higher Genus with Finite Total Curvature, Arch. Rational Mech. Anal. 137 (1997) 1–25