$h \rightarrow Z \gamma$ in the complex two Higgs doublet model

The latest LHC data confirmed the existence of a Higgs-like particle and made interesting measurements on its decays into $\gamma \gamma$, $Z Z^\ast$, $W W^\ast$, $\tau^+ \tau^-$, and $b \bar{b}$. It is expected that a decay into $Z \gamma$ might be measured at the next LHC round, for which there already exists an upper bound. The Higgs-like particle could be a mixture of scalar with a relatively large component of pseudoscalar. We compute the decay of such a mixed state into $Z \gamma$, and we study its properties in the context of the complex two Higgs doublet model, analysing the effect of the current measurements on the four versions of this model. We show that a measurement of the $h \rightarrow Z \gamma$ rate at a level consistent with the SM can be used to place interesting constraints on the pseudoscalar component. We also comment on the issue of a wrong sign Yukawa coupling for the bottom in Type II models.