  The geometry of the quaternionic anti-de Sitter fibration is studied in
details. As a consequence, we obtain formulas for the horizontal Laplacian and
subelliptic heat kernel of the fibration. The heat kernel formula is explicit
enough to derive small time asymptotics. Related twistor spaces and
corresponding heat kernels are also discussed and the connection to the
quaternionic magnetic Laplacian is done.
