  In 1900, Macfarlane proposed a hyperbolic variation on Hamilton's quaternions
that closely resembles Minkowski spacetime. Viewing this in a modern context,
we expand upon Macfarlane's idea and develop a model for real hyperbolic
3-space in which both points and isometries are expressed as complex
quaternions, analogous to Hamilton's famous theorem on Euclidean rotations. We
use this to give new computational tools for studying isometries of hyperbolic
2- and 3-space. We also give a generalization to other quaternion algebras.
