  We start by presenting the current status of a complex flavour conserving
two-Higgs doublet model. We will focus on some very interesting scenarios where
unexpectedly the light Higgs couplings to leptons and to b-quarks can have a
large pseudoscalar component with a vanishing scalar component. Predictions for
the allowed parameter space at end of the next run with a total collected
luminosity of $300 \, fb^{-1}$ and $3000 \, fb^{-1}$ are also discussed. These
scenarios are not excluded by present data and most probably will survive the
next LHC run. However, a measurement of the mixing angle $\phi_\tau$, between
the scalar and pseudoscalar component of the 125 GeV Higgs, in the decay $h \to
\tau^+ \tau^-$ will be able to probe many of these scenarios, even with low
luminosity. Similarly, a measurement of $\phi_t$ in the vertex $\bar t t h$
could help to constrain the low $\tan \beta$ region in the Type I model.
