  We study the most popular scalar extension of the Standard Model, namely the
Two Higgs doublet model, extended by a complex triplet scalar (2HDMcT). Such
considering model with a very small vacuum expectation value, provides a
solution to the massive neutrinos through the so-called type II seesaw
mechanism. We show that the 2HDMcT enlarged parameter space allow for a rich
and interesting phenomenology compatible with current experimental constraints.
In this paper the 2HDMcT is subject to a detailed scrutiny. Indeed, a complete
set of tree level unitarity constraints on the coupling parameters of the
potential is determined, and the exact tree-level boundedness from below
constraints on these couplings are generated for all directions. We then
perform an extensive parameter scan in the 2HDMcT parameter space, delimited by
the above derived theoretical constraints as well as by experimental limits. We
find that an important triplet admixtures are still compatible with the Higgs
data and investigate which observables will allow to restrict the triplet
nature most effectively in the next runs of the LHC. Finally, we emphasize new
production and decay channels and their phenomenological relevance and
treatment at the LHC.
