  Testing the properties of the Higgs particle discovered at the LHC and
searching for new physics signals, are some of the most important tasks of
Particle Physics today. Current measurements of the Higgs couplings to fermions
and gauge bosons, seem consistent with the Standard Model, and when taken as a
function of the particle mass, should lay on a single line. However, in models
with an extended Higgs sector the diagonal Higgs couplings to up-quarks,
down-quarks and charged leptons, could lay on different lines, while
non-diagonal flavor-violating Higgs couplings could appear too. We describe
these possibilities within the context of multi-Higgs doublet models that
employ the Froggatt-Nielsen (FN) mechanism to generate the Yukawa hierarchies.
Furthermore, one of the doublets can be chosen to be of the inert type, which
provides a viable dark matter candidate. The mixing of the Higgs doublets with
the flavon field, can provide plenty of interesting signals, including: i)
small corrections to the couplings of the SM-like Higgs, ii) exotic signals
from the flavon fields, iii) new signatures from the heavy Higgs bosons. These
aspects are studied within a specific model with 3+1 Higgs doublets and a
singlet FN field. Constraints on the model are derived from the study of K and
D mixing and the Higgs search at the LHC. For last, the implications from the
latter aforementioned constraints to the FCNC top decay $t\to ch$ are presented
too.
