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.
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