  We study the problem of existence of geometric structures on compact complex
surfaces that are related to split quaternions. These structures, called
para-hypercomplex, para-hyperhermitian and para-hyperk\"ahler are analogs of
the hypercomplex, hyperhermitian and hyperk\"ahler structures in the definite
case. We show that a compact oriented 4-manifold carries a para-hyperk\"ahler
structure iff it has a metric of split signature together with two parallel,
orthogonal and null vector fields. Every compact complex surface admiting a
para-hyperhermitian structure has vanishing first Chern class and we show that,
unlike the definite case, many of these surfaces carry infinite dimensional
families of such structures. We provide also compact examples of complex
surfaces with para-hyperhermitian structures which are not locally conformally
para-hyperk\"ahler. Finally, we discuss the problem of non-existence of
para-hyperhermitian structures on Inoue surfaces of type $S^0$ and provide a
list of compact complex surfaces which could carry para-hypercomplex
structures.
