  We construct examples of 2-step Carnot groups related to quaternions and
study their fine structure and geometric properties. This involves the
Hamiltonian formalism, which is used to obtain explicit equations for geodesics
and the computation of the number of geodesics joining two different points on
these groups. We able to find the explicit lengths of geodesics. We present the
fundamental solutions of the Heat and sub-Laplace equations for these
anisotropic groups and obtain some estimates for them, which may be useful.
