  We propose a new superspace formulation for N=(4,4) conformal supergravity in
two dimensions. This is based on a geometry where the structure group of the
curved superspace is chosen to be SO(1,1) x SU(2)_L x SU(2)_R. The off-shell
supergravity multiplet possesses super-Weyl transformations generated by an
unconstrained real scalar superfield. The new supergravity formulation turns
out to be an extension of the minimal multiplet introduced in 1988 by Gates et.
al. and it allows the existence of various off-shell matter supermultiplets.
Covariant twisted-II and twisted-I multiplets respectively describe the field
strength of an Abelian vector multiplet and its prepotential. Moreover, we
introduce covariant bi-projective superfields. These define a large class of
matter multiplets coupled to 2D N=(4,4) conformal supergravity. They are the
analogue of the covariant projective superfields recently introduced for 4D and
5D matter-coupled supergravity but they differ by the fact that bi-projective
superfields are defined with the use of two CP^1 instead of one. We conclude by
giving a manifestly locally supersymmetric and super-Weyl invariant action
principle in bi-projective superspace.
