  The effects of $CP$-violation on the reported result by the LHC
$\mu^{Z\gamma}=2.2\pm 0.7$ for the $H\rightarrow Z\gamma$ signal strength are
studied in new physics models, where bounds on the real and absorptive parts of
the $CP$-violating form factor $h_3^{Z\gamma}$ are obtained, which are found to
be less than 1.02 GeV. The leading order SM contributions to the $H\rightarrow
Z\gamma$ decay are also revisited, whereas contributions to the $CP$-violating
form factor $h_3^{Z\gamma}$ from FCNC complex couplings mediated by the $Z$ and
$H$ bosons are reported. Using the current bounds on such a couplings, where
for these related to the Higgs boson are calculated to be or order $10^{-1}$.
We find that the FCNC contribution to $h_3^{Z\gamma}$ with top and charm quarks
in the loop is of order $10^{-6}$. A generic model with new quarks is also
studied. It is found a parameter space, which agrees with the
$H\rightarrow\gamma\gamma$ decay and the new $\mu^{Z\gamma}$ LHC result.
