  We study a $U(1) \times U(1)$ gauge theory discussing its vortex solutions
and supersymmetric extension. In our set-upon the dynamics of one of two
Abelian gauge fields is governed by a Maxwell term, the other by a Chern-Simons
term. The two sectors via a BF gauge field mixing and a Higgs portal term that
connects the two complex scalars. We also consider the supersymmetric version
of this system which allows to find for the bosonic sector BPS equations in
which an additional real scalar field enters into play. We study numerically
the field equations finding vortex solutions with both magnetic flux and
electric charge.
