  We study models with several $SU(2)$ scalar doublets where the inert doublets
have a non-minimal coupling to gravity and play the role of the inflaton. We
allow for this coupling to be complex, thereby introducing CP-violation -- a
necessary source of the baryon asymmetry -- in the Higgs-inflaton couplings. We
investigate the inflationary dynamics of the model and discuss how the
CP-violation of the model is imprinted on the particle asymmetries after
inflation in the hot big bang universe.
