I will talk about a work in progress in collaboration with Christian El Emam, whose final motivation is to study the geometry of surface group actions on SL(2, C) regarded as the homogeneous space SO(4,C)/SO(3, C). I will mainly focussed on some analytical issues related to equivariant immersions. We will introduce the notion of first and second fundamental forms for an immersion and prove that they are solutions of a complex version of the standard Gauss Codazzi equation in the hyperbolic setting. We will discuss how far this theory can be regarded as a complex version of the Anti de Sitter geometry . In particular we will introduce a notion of left/right Gauss maps for an immersion that extends the corresponding notions in the Anti de Sitter setting. In the final part of the talk we will introduce the notion of minimal immersion in this context and will try to give a general description of the embedding data of minimal surfaces in terms of holomorphic objects.