  We prove necessary and sufficient conditions for the $L^p$-convergence,
$p>1$, of the Biggins martingale with complex parameter in the supercritical
branching random walk. The results and their proofs are much more involved
(especially in the case $p\in (1,2)$) than those for the Biggins martingale
with real parameter. Our conditions are ultimate in the case $p\geq 2$ only.
