The Fence of Catenoids (or Translation Invariant Catenoid) is a versatile surface that allows many generalizations, giving opportunities to explore possibilities that are otherwise hard (or impossible) to achieve. Here we add a simple handle to it, increasing the genus of the quotient surface to 2:

In this case, the additional handles are placed through the symmetry axes of the catenoids. One can also put them though the other vertical symmetry plane of the surface:

In either case, there is a 1-parameter family of such surfaces (assuming maximal symmetry), requiring a 1-dimensional period problem to be solved. It is not clear to me whether the handle can be placed elsewhere, and whether the two types of surfaces can be deformed into each other.