  We study projectional properties of Poisson cut-out sets $E$ in non-Euclidean
spaces. In the first Heisenbeg group, endowed with the Kor\'anyi metric, we
show that the Hausdorff dimension of the vertical projection $\pi(E)$
(projection along the center of the Heisenberg group) almost surely equals
$\min\{2,dim_H(E)\}$ and that $\pi(E)$ has non-empty interior if $dim_H(E)>2$.
As a corollary, this allows us to determine the Hausdorff dimension of $E$ with
respect to the Euclidean metric in terms of its Heisenberg Hausdorff dimension
$dim_H (E)$.
  We also study projections in the one-point compactification of the Heisenberg
group, that is, the $3$-sphere $S^3$ endowed with the visual metric $d$
obtained by identifying $S^3$ with the boundary of the complex hyperbolic
plane. In $S^3$, we prove a projection result that holds simultaneously for all
radial projections (projections along so called "chains"). This shows that the
Poisson cut-outs in $S^3$ satisfy a strong version of the Marstrand's
projection theorem, without any exceptional directions.
