Quaternionic spinors and horospheres in 4-dimensional hyperbolic geometry

We give explicit bijective correspondences between three families of objects: certain pairs of quaternions, which we regard as spinors; certain flags in (1+4)-dimensional Minkowski space; and horospheres in 4-dimensional hyperbolic space decorated with certain pairs of spinorial directions. These correspondences generalise previous work of the first author, Penrose--Rindler, and Penner in lower dimensions, and use the description of 4-dimensional hyperbolic isometries via Clifford matrices studied by Ahlfors and others. We show that lambda lengths generalise to 4 dimensions, where they take quaternionic values, and are given by a certain bilinear form on quaternionic spinors. They satisfy a non-commutative Ptolemy equation, arising from quasi-Pl\"ucker relations in the Gel'fand--Retakh theory of noncommutative determinants. We also study various structures of geometric and topological interest that arise in the process.