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We discuss four off-shell N=4 D=1 supersymmetry transformations, theirassociated one-dimensional sigma-models and their mutual relations. They aregiven by I) the (4,4)_{lin} linear supermultiplet (supersymmetric extension ofR^4), II) the (3,4,1)_{lin} linear supermultiplet (supersymmetric extension ofR^3), III) the (3,4,1)_{nl} non-linear supermultiplet living on S^3 and IV) the(2,4,2)_{nl} non-linear supermultiplet living on S^2. The I -> II map is thesupersymmetric extension of the R^4 -> R^3 bilinear map, while the II -> IV mapis the supersymmetric extension of the S^3 -> S^2 first Hopf fibration. Therestrictions on the S^3, S^2 spheres are expressed in terms of thestereographic projections. The non-linear supermultiplets, whosesupertransformations are local differential polynomials, are not equivalent tothe linear supermultiplets with the same field content. The sigma-models aredetermined in terms of an unconstrained prepotential of the target coordinates.The Uniformization Problem requires solving an inverse problem for theprepotential. The basic features of the supersymmetric extension of the secondand third Hopf maps are briefly sketched. Finally, the Schur's lemma (i.e. thereal, complex or quaternionic property) is extended to all minimal linearsupermultiplets up to N<=8.

Supersymmetric Extension of Hopf Maps: N=4 sigma-models and the S^3 -> S^2 Fibration