Applying Ribaucour transformations, we construct two new 3-parameter families of complete surfaces, immersed in $H^3$, with constant mean curvature 1 and infinitely many embedded horosphere type ends. Each surface of the first family is locally associated to an Enneper cousin. It has one irregular end and infinitely many regular ends asymptotic to horospheres. The surfaces of the second family are locally associated to a catenoid cousin. Each surface of this family has infinitely many embedded horosphere type ends and one regular end with infinite total curvature.
"New constant mean curvature surfaces in the hyperbolic space." Illinois J. Math. 53 (1) 135 - 161, Spring 2009. https://doi.org/10.1215/ijm/1264170843