Are there any analogues or generalizations of elliptic functions for quaternions?

Question

As far as I understand, there is a result due to Jacobi, that there can not exist any generalization of elliptic functions to multiperiodic (i.e. at least three-periodic) single-valued single complex variable functions. I was thinking that maybe one has considered some generalisation of these functions to the quaternionic case. There surely are hypereliptic finctions which are four-periodic, but that's because they depend on two variables, although their connection to quaternions is unknown to me (and any reference would be warmly welcomed). Though I'm mainly interested whether there exist, say, five-periodic functions in quaternionic variable (or two complex variables for that matter) or a result forbidding the existence of such functions.