The phase diagram at zero temperature of a lattice SU(2) scalar-fermion model in 2+1 dimensions is studied numerically and with mean-field methods. Special attention is devoted to the strong coupling regime. We have developed a new method to adapt the hybrid Monte Carlo algorithm to the O(3) non-linear $\ensuremath{\sigma}$ model constraint. The charged excitations in the various phases are studied at the mean-field level. Bound states of two charged fermions are found in a strongly coupled paramagnetic phase. On the other hand, in the strongly coupled antiferromagnetic phase fermionic excitations around momenta $(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2)$ emerge.