  We present a detailed study of the annihilation signals of the inert dark
matter doublet model in its high mass regime. Concretely, we study the
prospects to observe gamma-ray signals of the model in current and projected
Cherenkov telescopes taking into account the Sommerfeld effect and including
the contribution to the spectrum from gamma-ray lines as well as from internal
bremsstrahlung. We show that present observations of the galactic center by the
H.E.S.S. instrument are able to exclude regions of the parameter space that
give the correct dark matter relic abundance. In particular, models with the
charged and the neutral components of the inert doublet nearly degenerate in
mass have strong gamma-ray signals. Furthermore, for dark matter particle
masses above 1 TeV, we find that the non-observation of the continuum of
photons generated by the hadronization of the annihilation products typically
gives stronger constraints on the model parameters than the sharp spectral
features associated to annihilation into monochromatic photons and the internal
bremsstrahlung process. Lastly, we also analyze the interplay between indirect
and direct detection searches for this model, concluding that the prospects for
the former are more promising. In particular, we find that the upcoming
Cherenkov Telescope Array will be able to probe a significant part of the high
mass regime of the model.
