  We propose a new type of search for a pseudoscalar particle $\eta$ pair
produced via an off-shell Higgs, $pp\to h^*\to \eta\eta$. The search is
motivated by a composite Higgs model in which the $\eta$ is extremely narrow
and decays almost exclusively into $Z\gamma$ in the mass range $65$
GeV$\lesssim m_\eta \lesssim 160$ GeV. We devise an analysis strategy to
observe the novel $Z\gamma Z\gamma$ channel and estimate potential bounds on
the Higgs-$\eta$ coupling. The experimental sensitivity to the signatures
depends on the power to identify fake photons and on the ability to predict
large photon multiplicities. This search allows us to exclude large values of
the compositeness scale $f$, being thus complementary to other typical
processes.
