  We study the geometry of the Margulis region associated with an irrational
screw translation $g$ acting on the 4-dimensional real hyperbolic space. This
is an invariant domain with the parabolic fixed point of $g$ on its boundary
which plays the role of an invariant horoball for a translation in dimensions
$\leq 3$. The boundary of the Margulis region is described in terms of a
function $B_\alpha : [0,\infty) \to {\mathbb R}$ which solely depends on the
rotation angle $\alpha \in {\mathbb R}/{\mathbb Z}$ of $g$. We obtain an
asymptotically universal upper bound for $B_\alpha(r)$ as $r \to \infty$ for
arbitrary irrational $\alpha$, as well as lower bounds when $\alpha$ is
Diophatine and the optimal bound when $\alpha$ is of bounded type. We
investigate the implications of these results for the geometry of Margulis
cusps of hyperbolic 4-manifolds that correspond to irrational screw
translations acting on the universal cover. Among other things, we prove
bi-Lipschitz rigidity of these cusps.
