We study geometric structures on compact complex surfaces that are related to split quaternions. These structures, called para-hypercomplex, para-hyperhermitian and para-hyperkaehler (or hypersymplectic) , are analogs of the hypercomplex, hyperhermitian and hyperkaehler structures in the definite case. Every compact complex surface admitting a para-hyperhermitian structure has vanishing first Chern class and, unlike the definite case, many of these surfaces carry infinite dimensional families of such structures. In the talk we´ll consider the relation among the different structures together with several examples and some open problems.