We discuss four off-shell N=4 D=1 supersymmetry transformations, their associated one-dimensional sigma-models and their mutual relations. They are given by I) the (4,4)_lin linear supermultiplet (supersymmetric extension of R^4), II) the (3,4,1)_lin linear supermultiplet (supersymmetric extension of R^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 the supersymmetric extension of the R^4 -> R^3

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... linear map, while the II -> IV map is the supersymmetric extension of the S^3 -> S^2 first Hopf fibration. The restrictions on the S^3, S^2 spheres are expressed in terms of the stereographic projections. The non-linear supermultiplets, whose supertransformations are local differential polynomials, are not equivalent to the linear supermultiplets with the same field content. The sigma-models are determined in terms of an unconstrained prepotential of the target coordinates. The Uniformization Problem requires solving an inverse problem for the prepotential. The basic features of the supersymmetric extension of the second and third Hopf maps are briefly sketched. Finally, the Schur's lemma (i.e. the real, complex or quaternionic property) is extended to all minimal linear supermultiplets up to N<=8.