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We consider a multi-component dark matter model where the dark sectorcontains 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. Interactionsbetween the dark sectors impact significantly dark matter observables, theyallow in particular to significantly relax the direct detection constraints onthe model. To determine the parameter space that satisfies relic density,theoretical and collider constraints as well as direct and indirect detectionlimits, we perform two separate scans, the first includes the full parameterspace of the model while the second is dedicated to scenarios with a compressedinert doublet spectrum. In the first case we find that the singlet is generallythe dominant dark matter component while in the compressed case the doublet ismore likely to be the dominant dark matter component. In both cases we findthat 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 cosmologicalastrophysical and collider constraints and show that a large fraction of theparameter space that escapes current constraints is within the sensitivityreach of future detectors such as XENON-nT, Darwin or CTA. Important collidersignatures are mostly found in the compressed spectrum case with thepossibility of probing the model with searches for heavy stable chargedparticles and disappearing tracks. We also show that semi-annihilationprocesses such as $\tilde{S}\tilde{S}\to \tilde{A}Z$ could give the dominantsignature in indirect detection searches.

Two dark matter candidates: the case of inert doublet and singlet scalars