  We consider a multi-component dark matter model where the dark sector
contains a scalar doublet and a complex scalar singlet. We impose a discrete
$Z_4$ symmetry to ensure such that the lightest component of the doublet,
$\tilde{A}$, and the singlet, $\tilde{S}$, are both stable. Interactions
between the dark sectors impact significantly dark matter observables, they
allow in particular to significantly relax the direct detection constraints on
the model. To determine the parameter space that satisfies relic density,
theoretical and collider constraints as well as direct and indirect detection
limits, we perform two separate scans, the first includes the full parameter
space of the model while the second is dedicated to scenarios with a compressed
inert doublet spectrum. In the first case we find that the singlet is generally
the dominant dark matter component while in the compressed case the doublet is
more likely to be the dominant dark matter component. In both cases we find
that the two dark matter particles can have masses that ranges from around
$m_h/2$ to over the TeV scale. We emphasize the interplay between cosmological
astrophysical and collider constraints and show that a large fraction of the
parameter space that escapes current constraints is within the sensitivity
reach of future detectors such as XENON-nT, Darwin or CTA. Important collider
signatures are mostly found in the compressed spectrum case with the
possibility of probing the model with searches for heavy stable charged
particles and disappearing tracks. We also show that semi-annihilation
processes such as $\tilde{S}\tilde{S}\to \tilde{A}Z$ could give the dominant
signature in indirect detection searches.
