  This work begins with a review of complexification and realification of Hopf
algebras. We emphasize the notion of multiplier Hopf algebras for the
description of different classes of functions (compact supported, bounded,
unbounded) on complex quantum groups and the construction of the associated
left and right Haar measure. Using a continuation of $6j$ symbols of $SU_q (2)$
with complex spins, we give a new description of the unitary representations of
$SL_q (2,\CC)_{\RR}$ and find explicit expressions for the characters of $SL_q
(2,\CC)_{\RR}$. The major theorem of this article is the Plancherel theorem for
the Quantum Lorentz Group.
