  We make a phenomenological study of a model with two inert doublets plus one
Higgs doublet (I(2+1)HDM) which is symmetric under a Z$_2$ group, preserved
after Electro-Weak Symmetry Breaking (EWSB) by the vacuum alignment $(0,0,v)$.
This model may be regarded as an extension to the model with one inert doublet
plus one Higgs doublet (I(1+1)HDM), by the addition of an extra inert scalar
doublet. The neutral fields from the two inert doublets provide a viable Dark
Matter (DM) candidate which is stabilised by the conserved $Z_2$ symmetry. We
study the new Higgs decay channels offered by the scalar fields from the extra
doublets and their effect on the Standard Model (SM) Higgs couplings, including
a new decay channel into photon(s) plus missing energy, which distinguishes the
I(2+1)HDM from the I(1+1)HDM. Motivated by Supersymmetry, which requires an
even number of doublets, we then extend this model into a model with four inert
doublets plus two Higgs doublets (I(4+2)HDM) and study the phenomenology of the
model with the vacuum alignment $(0,0,0,0,v,v)$. This scenario offers a wealth
of Higgs signals, the most distinctive ones being cascade decays of heavy Higgs
states into inert ones. Finally, we also remark that the smoking-gun signature
of all the considered models is represented by invisible Higgs decays into the
lightest inert Higgs bosons responsible for DM.
