  The 2-body problem on the sphere and hyperbolic space are both real forms of
holomorphic Hamiltonian systems defined on the complex sphere. This admits a
natural description in terms of biquaternions and allows us to address
questions concerning the hyperbolic system by complexifying it and treating it
as the complexification of a spherical system. In this way, results for the
2-body problem on the sphere are readily translated to the hyperbolic case. For
instance, we implement this idea to completely classify the relative equilibria
for the 2-body problem on hyperbolic 3-space for a strictly attractive
potential.
