  We consider the Higgs inflation in the extension of the Standard Model with
two Higgs doublets coupled to gravity non-minimally. In the presence of an
approximate global U(1) symmetry in the Higgs sector, both radial and angular
modes of neutral Higgs bosons drive inflation where large non-Gaussianity is
possible from appropriate initial conditions on the angular mode. We also
discuss the case with single-field inflation for which the U(1) symmetry is
broken to a Z_2 subgroup. We show that inflationary constraints, perturbativity
and stability conditions restrict the parameter space of the Higgs quartic
couplings at low energy in both multi- and single-field cases. Focusing on the
inert doublet models where Z_2 symmetry remains unbroken at low energy, we show
that the extra neutral Higgs boson can be a dark matter candidate consistent
with the inflationary constraints. The doublet dark matter is always heavy in
multi-field inflation while it can be light due to the suppression of the
co-annihilation in single-field inflation. The implication of the extra quartic
couplings on the vacuum stability bound is also discussed in the light of the
recent LHC limits on the Higgs mass.
