  We study the case of a pseudo-scalar dark matter candidate which emerges from
a complex scalar singlet, charged under a global U(1) symmetry, which is broken
both explicitly and spontaneously. The pseudo-scalar is naturally stabilized by
the presence of a remnant discrete symmetry: dark CP. We study and compare the
phenomenology of several simplified models with only one explicit symmetry
breaking term. We find that several regions of the parameter space are able to
reproduce the observed dark matter abundance while respecting direct detection
and invisible Higgs decay limits: in the resonances of the two scalars,
featuring the known as forbidden or secluded dark matter, and through
non-resonant Higgs-mediated annihilations. In some cases, combining different
measurements would allow one to distinguish the breaking pattern of the
symmetry. Moreover, this setup admits a light DM candidate at the sub-GeV
scale. We also discuss the situation where more than one symmetry breaking term
is present. In that case, the dark CP symmetry may be spontaneously broken,
thus spoiling the stability of the dark matter candidate. Requiring that this
does not happen imposes a constraint on the allowed parameter space. Finally,
we consider an effective field theory approach valid in the
pseudo-Nambu-Goldstone boson limit and when the U(1) breaking scale is much
larger than the electroweak scale.
