  We discuss the CP nature of the Yukawa couplings of the Higgs boson in the
framework of a complex two Higgs doublet model (C2HDM). After analysing all
data gathered during the Large Hadron Collider run 1, the measurement of the
Higgs couplings to the remaining SM particles already restricts the parameter
space of many extensions of the SM. However, there is still room for very large
CP-odd Yukawa couplings to light quarks and leptons while the top-quark Yukawa
coupling is already very constrained by current data. Although indirect
measurements of electric dipole moments play a very important role in
constraining the pseudo-scalar components of the Yukawa couplings, we argue
that a direct measurement of the ratio of pseudoscalar to scalar couplings
should be one of the top priorities for the LHC run 2.
