A quasi-Fuchsian representation of a surface group in PSL(2,C) is a discrete and faithful representation that preserves a Jordan curve on the Riemann sphere. These classical objects have a very rich structure as they lie at the crossroad of several areas of mathematics such as complex dynamics, Teichmüller theory, and 3-dimensional hyperbolic geometry. I will discuss joint work with James Farre and Gabriele Viaggi in which we investigate similar phenomena for a class of representations of surface groups in PSL(d,C), hyperconvex representations, and discuss geometric properties of the image subgroups and their parameter space. Among other things we show that the groups admitting hyperconvex  representations are virtually isomorphic to convex-cocompact subgroups of PSL(2,C), and more generally they exhibit striking analogies with such groups, such as suitable Ahlfors-Bers parameters.