  Biggins [Uniform convergence of martingales in the branching random walk.
{\em Ann. Probab.}, 20(1):137--151, 1992] proved local uniform convergence of
additive martingales in $d$-dimensional supercritical branching random walks at
complex parameters $\lambda$ from an open set $\Lambda \subseteq \mathbb{C}^d$.
We investigate the martingales corresponding to parameters from the boundary
$\partial \Lambda$ of $\Lambda$. The boundary can be decomposed into several
parts. There may be a part of the boundary, on which the martingales do not
exist, on other parts it exists, but diverges or vanishes in the limit. In the
remaining part, there is convergence to a non-degenerate limit. The arguments
that give this convergence also apply in $\Lambda$ and require weaker moment
assumptions than the ones used by Biggins.
