  We study the geometry of hyperconvex representations of surface groups in
${\rm PSL}(d,\mathbb{C})$ and their deformation spaces: We produce a natural
holomorphic extension of the classical Ahlfors--Bers map to a product of
Teichm\"uller spaces of a canonical Riemann surface lamination and prove that
the limit set of a hyperconvex representation in the full flag space has
Hausdorff dimension 1 if and only if the representation is conjugate in ${\rm
PSL}(d,\mathbb{R})$.
