  In this paper we introduce a discrete integrable system generalizing the
discrete (real) cross-ratio system in $S^4$ to complex values of a generalized
cross-ratio by considering $S^4$ as a real section of the complex Pl\"ucker
quadric, realized as the space of two-spheres in $S^4.$ We develop the geometry
of the Pl\"ucker quadric by examining the novel contact properties of
two-spheres in $S^4,$ generalizing classical Lie geometry in $S^3.$ Discrete
differential geometry aims to develop discrete equivalents of the geometric
notions and methods of classical differential geometry. We define discrete
principal contact element nets for the Pl\"ucker quadric and prove several
elementary results. Employing a second real real structure, we show that these
results generalize previous results by Bobenko and Suris $(2007)$ on discrete
differential geometry in the Lie quadric.
