  We develop a way of seeing a complete orientable hyperbolic $4$-manifold
$\mathcal{M}$ as an orbifold cover of a Coxeter polytope $\mathcal{P} \subset
\mathbb{H}^4$ that has a facet colouring. We also develop a way of finding
totally geodesic sub-manifolds $\mathcal{N}$ in $\mathcal{M}$, and describing
the result of mutations along $\mathcal{N}$. As an application of our method,
we construct an example of a complete orientable hyperbolic $4$-manifold
$\mathcal{X}$ with a single non-toric cusp and a complete orientable hyperbolic
$4$-manifold $\mathcal{Y}$ with a single toric cusp. Both $\mathcal{X}$ and
$\mathcal{Y}$ have twice the minimal volume among all complete orientable
hyperbolic $4$-manifolds.
