  An analysis of the off-shell $H^\ast\rightarrow ZZ \rightarrow
\overline{\ell}_1\ell_1\overline{\ell}_2\ell_2$ decay width is presented for
both unpolarized and polarized $Z$ gauge bosons in the scenario with the most
general $H^*ZZ$ vertex function, which is given in terms of two $CP$-even
($\hat b_Z$ and $\hat c_Z$) and one $CP$-odd ($\tilde b_Z$) anomalous
couplings.
  The SM contributions to the $H^*ZZ$ coupling up to the one-loop level are
also included. Explicit analytic results for the unpolarized and polarized
$H^\ast\rightarrow ZZ \rightarrow
\overline{\ell}_1\ell_1\overline{\ell}_2\ell_2$ square amplitudes and the
four-body phase space are presented, out of which several observable quantities
can be obtained straightforwardly. As far as the numerical analysis is
concerned, a cross-check is performed via \texttt{MadGraph5\_aMC@NLO}, where
our model was implemented with the aid of FeynRules. We then consider the most
stringent bounds on anomalous complex $H^*ZZ$ couplings and analyze the effects
of the polarizations of the $Z$ gauge bosons through the polarized
$H^\ast\rightarrow ZZ \rightarrow
\overline{\ell}_1\ell_1\overline{\ell}_2\ell_2$ decay width as well as
left-right and forward-backward asymmetries, which are found to be sensitive to
new-physics effects. Particular focus is put on the effects of the absorptive
parts of the anomalous $H^*ZZ$ couplings, which have been largely overlooked up
to now in LHC analyses. It is found that the studied observable quantities,
particularly the left-right asymmetries, can be helpful to look for effects of
$CP$-violation in the $H^*ZZ$ coupling and set bounds on the absorptive parts.
For completeness, we also analyze the case of unpolarized $Z$ gauge bosons.
