  We study a new class of renormalisable simplified models for dark matter
searches at the LHC that are based on two Higgs doublet models with an
additional pseudoscalar mediator. In contrast to the spin-0 simplified models
employed in analyses of Run I data these models are self-consistent, unitary
and bounds from Higgs physics typically pose no constraints. Predictions for
various missing transverse energy ($E_{T, \rm miss}$) searches are discussed
and the reach of the 13 TeV LHC is explored. It is found that the proposed
models provide a rich spectrum of complementary observables that lead to
non-trivial constraints. We emphasise in this context the sensitivity of the
$t\bar t + E_{T, \rm miss}$, mono-$Z$ and mono-Higgs channels, which yield
stronger limits than mono-jet searches in large parts of the parameter space.
Constraints from spin-0 resonance searches, electroweak precision measurements
and flavour observables are also derived and shown to provide further important
handles to constraint and to test the considered dark matter models.
