The Gosper island tiles the plane, so I'm curious if a nontrivial elliptic? function exists which would have a 'period gosperisland' instead of a period parallelogram. In this case, I'm using 'trivial' to denote a function that's got the right kind of lattice, but the Gosper island is overkill and unnecessary  for example the derivative of the equianharmonic case: $d\wp(z;0,1)/dz$ (last image).

The Gosper islands are a fundamental domain for the translation action of the Eisenstein integers $\mathbb{Z}[\frac{1+\sqrt{3}}{2}]$, since the shapes can be constructed by deforming a Voronoi decomposition of the plane with respect to that lattice. It is therefore reasonable to look for functions in the field generated by the Weierstrass $\wp$ function for the Eisenstein integers and its first derivative $\wp'$, since this field comprises all of the meromorphic functions that are periodic with respect to the lattice. It is not clear what selection rule you want to apply to favor one function over another. Elliptic functions do not have canonical fundamental domains, and one has to choose extra data (e.g., a basis of the lattice, and a pair of paths in the homotopy class representing the basis) to write down boundaries in the usual theory. I suppose you may want to find a function $f(z)$ such that the boundary configuration is equal to $\{ z : f(z) = 1 \}$, but I am somewhat doubtful that such a function exists, simply by degree considerations. 

