Self-dual vortex solutions are studied in detail in the generalized Abelian Higgs model with an 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 CP1 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 CP1 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.