  Modelled on a real hypersurface in a quaternionic manifold, we introduce a
quaternionic analogue of CR structure, called quaternionic CR structure. We
define the strong pseudoconvexity of this structure as well as the notion of
quaternionic pseudohermitian structure. Following the construction of the
Tanaka-Webster connection in complex CR geometry, we construct a canonical
connection associated with a quaternionic pseudohermitian structure, when the
underlying quaternionic CR structure satisfies the ultra-pseudoconvexity which
is stronger than the strong pseudoconvexity. Comparison to Biquard's
quaternionic contact structure is also made.
