  We study the geometry of oriented right-angled hexagons in H^4, the
hyperbolic 4-space, via Clifford numbers or quaternions. We show how to augment
alternate sides of such a hexagon so that for the non-augmented sides, we can
define quaternion half side-lengths whose angular parts are obtained from half
the Euler angles associated to a certain orientation-preserving isometry of the
Euclidean 3-space. This generalizes the complex half side-lengths of oriented
right-angled hexagons in H^3. We also define appropriate complex half
side-lengths for the augmented sides of the hexagon. We further explain how to
geometrically read off the quaternion half side-lengths for a given
oriented,augmented, right-angled hexagon in H^4. Our main result is a set of
generalized Delambre-Gauss formulas for oriented, augmented, right-angled
hexagons in H^4, involving the quaternion half side-lengths and the complex
half side-lengths. We also show in the appendix how the same method gives
Delambre-Gauss formulas for oriented right-angled hexagons in H^3, from which
the well-known sine and cosine laws can be deduced. These formulas generalize
the classical Delambre-Gauss formulas for spherical/hyperbolic triangles.
