  We present a detailed study of the most general N=2 supersymmetric sigma
models in four-dimensional anti-de Sitter space AdS_4 formulated in terms of
N=1 chiral superfields. The target space is demonstrated to be a non-compact
hyperkahler manifold restricted to possess a special Killing vector field which
generates an SO(2) group of rotations on the two-sphere of complex structures
and necessarily leaves one of them invariant. All hyperkahler cones, that is
the target spaces of N=2 superconformal sigma models, prove to possess such a
vector field that belongs to the Lie algebra of an isometry group SU(2) acting
by rotations on the complex structures. A unique property of the N=2 sigma
models constructed is that the algebra of OSp(2|4) transformations closes off
the mass shell. We uncover the underlying N=2 superfield formulation for the
N=2 sigma models constructed and compute the associated N=2 supercurrent.
  We give a special analysis of the most general systems of self-interacting
N=2 tensor multiplets in AdS_4 and their dual sigma models realized in terms of
N=1 chiral multiplets. We also briefly discuss the relationship between our
results on N=2 supersymmetric sigma models formulated in the N=1 AdS superspace
and the off-shell sigma models constructed in the N=2 AdS superspace in
arXiv:0807.3368.
