  We study an extension of the Standard Model (SM) in which two copies of the
SM Higgs doublet are added to the scalar sector. These extra doublets do not
develop a vacuum expectation value, hence, they are \textit{inert}. This
essentially leads to a 3-Higgs Doublet Model (3HDM) with 2 inert and 1 active
scalar doublets, which we denote as I(2+1)HDM. We allow for CP-violation in the
\textit{inert} sector, where the lightest \textit{inert} state is protected
from decaying to SM particles through the conservation of a $Z_2$ symmetry, so
that it is a Dark Matter (DM) candidate. For this scenario, we identify a
smoking gun signature of dark CP-violation in the form of production thresholds
of pairs of \textit{inert} neutral Higgs bosons at an $e^+e^-$ collider.
