  We give a new proof of the classification of contact real hypersurfaces with
constant mean curvature in the complex hyperbolic quadric ${Q^m}^* =
SO_{m,2}^o/SO_mSO_2$, where $m\geq 3$. We show that a contact real hypersurface
$M$ in ${Q^m}^*$ for $m\geq 3$ is locally congruent to a tube of radius
$r{\in}{\mathbb R}^+$ around the complex hyperbolic quadric ${Q^{m-1}}^*$, or
to a tube of radius $r\in\mathbb{R}^+$ around the $\mathfrak A$-principal
$m$-dimensional real hyperbolic space ${\mathbb R}H^m$ in ${Q^m}^* =
SO_{m,2}^o/SO_mSO_2$, or to a horosphere in ${Q^{m-1}}^*$ induced by a class of
$\mathfrak A$-principal geodesics in ${Q^m}^*$.
