  Proceeding from the superfield action for $N=4, d=1$ nonlinear
supermultiplet, equipped with the most general potential term, we find the
action describing a charged particle on the sphere $S^3$ in the field of $n$
fixed Dirac dyons. We construct the supercharges and Hamiltonian and analyze
some particulary interesting potentials corresponding to the N=4 supersymmetric
extension of the integrable one- and two-center
McIntosh--Cisneros--Zwanziger--Kepler (MICZ-Kepler) systems on $S^3$.
