  We investigate and compare two simple models of dark matter (DM): a vector
and a scalar DM model. Both models require the presence of two physical Higgs
bosons $h_1$ and $h_2$ which come from mixed components of the standard Higgs
doublet $H$ and a complex singlet $S$. In the Vector model, the extra $U(1)$
symmetry is spontaneously broken by the vacuum of the complex field $S$. This
leads to a massive gauge boson $X^\mu$ that is a DM candidate stabilized by the
dark charge conjugation symmetry $S \to S^*$, $X^\mu\to -X^\mu$. On the other
hand, in the Scalar model the gauge group remains the standard one. The DM
field $A$ is the imaginary component of $S$ and the stabilizing symmetry is
also the dark charge conjugation $S \to S^*$ ($A \to - A$). In this case, in
order to avoid spontaneous breaking, the $U(1)$ symmetry is broken explicitly,
but softly, in the scalar potential. The possibility to disentangle the two
models has been investigated. We have analyzed collider, cosmological, DM
direct and indirect detection constraints and shown that there are regions in
the space spanned by the mass of the non-standard Higgs boson and the mass of
the DM particle where the experimental bounds exclude one of the models. We
have also considered possibility to disentangle the models at $e^+e^-$ collider
and concluded that the process $e^+e^-\to Z + \text{DM}$ provides a useful tool
to distinguish the models.
