  The significance of new physics appearing in the loop-induced decays of
neutral Higgs bosons into pairs of dibosons $\gamma\gamma$ and $Z\gamma$ will
be discussed in the framework of the 3-3-1 models based on a recent
work~\cite{Okada:2016whh}, where the Higgs sector becomes effectively the same
as that in the two Higgs doublet models (2HDM) after the first symmetry
breaking from $SU(3)_L$ scale into the electroweak scale. For large $SU(3)_L$
scale $v_3\simeq10$ TeV, dominant one-loop contributions to the two decay
amplitudes arise from only the single charged Higgs boson predicted by the
2HDM, leading to that experimental constraint on the signal strength
$\mu^{331}_{\gamma\gamma}$ of the Standard Model-like Higgs boson decay
$h\rightarrow \gamma\gamma$ will result in a strict upper bound on the signal
strength $\mu^{331}_{Z\gamma}$ of the decay $h\rightarrow\, Z\gamma$. For a
particular model with lower $v_3$ around 3 TeV, contributions from heavy
charged gauge and Higgs bosons may have the same order, therefore may give
strong destructive or constructive correlations. As a by-product, a deviation
from the SM prediction $|\mu^{331}_{\gamma\gamma}-1| \le 0.04$ still allows
$|\mu^{331}_{Z\gamma}-1|$ to reach values near 0.1. We also show that there
exists an $CP$-even neutral Higgs boson $h^0_3$ predicted by the 3-3-1 models,
but beyond the 2HDM, has an interesting property that the branching ratio
Br$(h^0_3\rightarrow \gamma\gamma)$ is very sensitive to the parameter $\beta$
used to distinguish different 3-3-1 models.
