Conformal-biharmonic maps arise as critical points of the conformal-bienergy functional, a second-order functional obtained by modifying the classical bienergy functional through curvature terms motivated by conformal geometry. In dimension four, this functional enjoys conformal invariance, making it particularly relevant in the study of variational problems in conformal geometry. In this talk we focus on stability properties of conformal-biharmonic maps. We investigate the conformal-biharmonic stability of the identity map of compact Einstein manifolds with non-negative scalar curvature and show that its conformal-biharmonic index coincides with the classical harmonic index, with a remarkable exception in the case of the four-dimensional sphere. We also study conformal-biharmonic hypersurfaces in space forms and compute the index and nullity for hyperspheres in spheres. Our results reveal new stability phenomena and emphasize the geometric differences between biharmonic and conformal-biharmonic variational theories.