This surface owes its existence a typographical mistake I made in a Mathematica notebook.
This surface is clearly not embedded, but the self-intersections occur along straight lines, and the angle between the two tangent planes at the intersection is bounded away from zero. So maybe somebody ambitious can desingularize this by squeezing in Scherk surfaces?
Besides that, the surface deserves mild interest because it is related to Scherk’s fourth surface, and because its Weierstrass representation is very simple:
The conjugate surface could be called half a fence of catenoids.