  The Higgs boson discovered at the LHC opened a new chapter for particle
physics. Its properties need to be studied in detail to distinguish a purely
standard model (SM) Higgs boson from one of many scalars in an enlarged Higgs
sector. The CMS collaboration has reported a possible lepton flavor violating
(LFV) signal $h\to\mu\tau$, which if confirmed, implies that the Higgs sector
is larger than in the SM. New physics responsible for this type of decay may,
in general, also introduce other observable effects such as charge-parity (CP)
violation in $h\to \tau\tau$. We study two types of models that single out the
third generation and can induce large $h \to \mu\tau$ rates with different
consequences for CP violation in $h \to \tau \tau$. Predictions for the size of
the CP violating couplings require knowledge of the lepton Yukawa matrices and
we discuss this in the context of two different textures considering all
existing constraints.
