  Self-dual vortex solutions are studied in detail in the generalized abelian
Higgs model with independent Chern-Simons interaction. For special choices of
couplings, it reduces to a Maxwell-Higgs model with two scalar fields, a
Chern-Simons-Higgs model with two scalar fields, or other new models. We
investigate the properties of the static solutions and perform detailed
numerical analyses. For the Chern-Simons-Higgs model with two scalar fields in
an asymmetric phase, we prove the existence of multisoliton solutions which can
be viewed as hybrids of Chern-Simons vortices and $CP^1$ lumps. We also discuss
solutions in a symmetric phase with the help of the corresponding exact
solutions in its nonrelativistic limit. The model interpolating all three
models---Maxwell-Higgs, Chern-Simons-Higgs, and $CP^1$ models--- is discussed
briefly. Finally we study the possibility of vortex solutions with half-integer
vorticity in the special case of the model. Numerical results are negative.
