We consider the Sp(2n) invariant formulation of higher spin fields on flat and curved backgrounds of constant curvature. In this formulation an infinite number of higher spin fields are packed into single scalar and spinor master fields (hyperfields) propagating on extended spaces, to be called hyperspaces, parametrized by tensorial coordinates. We show that the free field equations on flat and AdS-like hyperspaces are related to each other by a generalized conformal transformation of the scalar and spinor master fields. We compute the four-point functions on a flat hyperspace for both scalar and spinor master fields, thus extending the two- and three-point function results of hep-th/0312244. Then using the generalized conformal transformation we derive two-, three- and four-point functions on AdS-like hyperspace from the corresponding correlators on the flat hyperspace.