  We investigate a possible realization of pseudo-Nambu-Goldstone (pNG) dark
matter in the framework of a singlet-extended 2 Higgs doublet model (S2HDM).
pNG dark matter gained attraction due to the fact that direct-detection
constraints can be avoided naturally because of the momentum-suppressed
scattering cross sections, whereas the relic abundance of dark matter can
nevertheless be accounted for via the usual thermal freeze-out mechanism. We
confront the S2HDM with a multitude of theoretical and experimental
constraints, paying special attention to the theoretical limitations on the
scalar potential, such as vacuum stability and perturbativity. In addition, we
discuss the complementarity between constraints related to the dark matter
sector, on the one hand, and to the Higgs sector, on the other hand. In our
numerical discussion we explore the Higgs funnel region with dark matter masses
around $60$GeV using a genetic algorithm. We demonstrate that the S2HDM can
account for the measured relic abundance while being in agreement with all
relevant constraints. We also discuss whether the so-called galactic center
excess and the antiproton excess can be accommodated, possibly in combination
with a Higgs boson at about $96$GeV that can be the origin of the LEP- and the
CMS-excess observed at this mass in the $b \bar b$-quark and the diphoton final
state, respectively.
