  We introduce and study some deformations of complete finite-volume hyperbolic
four-manifolds that may be interpreted as four-dimensional analogues of
Thurston's hyperbolic Dehn filling.
  We construct in particular an analytic path of complete, finite-volume cone
four-manifolds $M_t$ that interpolates between two hyperbolic four-manifolds
$M_0$ and $M_1$ with the same volume $\frac {8}3\pi^2$. The deformation looks
like the familiar hyperbolic Dehn filling paths that occur in dimension three,
where the cone angle of a core simple closed geodesic varies monotonically from
$0$ to $2\pi$. Here, the singularity of $M_t$ is an immersed geodesic surface
whose cone angles also vary monotonically from $0$ to $2\pi$. When a cone angle
tends to $0$ a small core surface (a torus or Klein bottle) is drilled
producing a new cusp.
  We show that various instances of hyperbolic Dehn fillings may arise,
including one case where a degeneration occurs when the cone angles tend to
$2\pi$, like in the famous figure-eight knot complement example.
  The construction makes an essential use of a family of four-dimensional
deforming hyperbolic polytopes recently discovered by Kerckhoff and Storm.
