Building on the five-dimensional constructions in hep-th/0601177, we provide a unified description of four-dimensional = 2 superconformal off-shell multiplets in projective superspace, including a realization in terms of = 1 superfields. In particular, superconformal polar multiplets are consistently defined for the first time. We present new 4D = 2 superconformal sigma-models described by polar multiplets. Such sigma-models realize general superconformal couplings in projective superspace, but involve an infinite tale of auxiliary = 1 superfields. The auxiliaries should be eliminated by solving infinitely many algebraic nonlinear equations, and this is a nontrivial technical problem. We argue that the latter can be avoided by making use of supersymmetry considerations. All information about the resulting superconformal model (and hence the associated hyperkähler cone) is encoded in the so-called canonical coordinate system for a Kähler metric, which was introduced by Bochner and Calabi in the late 1940s.