In this paper we describe the moduli spaces of degree d branched superminim
al immersions of a compact Riemann surface of genus g into S-4. We prove th
at when d greater than or equal to max{2g, g + 2}, such spaces have the str
ucture of projectivized fibre products and are path-connected quasi-project
ive varieties of dimension 2d - g + 4. This generalizes known results for s
paces of harmonic 2-spheres in S-4.