  Superconformal matter multiplets play a crucial role in the construction of
Poincare supergravity invariants. Off-shell multiplets allow for construction
of general matter couplings in supergravity. In Nucl. Phys. B214 (1983)
519-531, relaxed hypermultiplet was constructed in rigid supersymmetry which on
coupling with the real scalar multiplet allowed for an off-shell formulation of
the rigid hypermultiplet. In this paper, we extend the relaxed hypermultiplet
to conformal supergravity. For consistency with the superconformal algebra, we
find that the fields have to be allowed to transform in a non-canonical way
under SU(2) symmetry. We find suitable field redefinitions to obtain fields
which are irreducible representations of SU(2) R-symmetry and present the full
non-linear transformation rule.
