We elaborate on a novel superconformal mechanics model possessing D(2, 1; alpha) symmetry and involving extra U(2) spin variables. It is the one-particle case of the N=4 superconformal matrix model recently proposed in arXiv:0812.4276 [hep-th], and it generalizes to arbitrary a alpha not equal 0 the OSp(4|2) superconformal mechanics of arXiv:0905.4951 [hep-th]. As in the latter case, the U(2) spin variables are described by a Wess-Zumino action and define the first Hopf map S-3 -> S-2 in the target space. Upon quantization, they represent a fuzzy sphere. We find the classical and quantum generators of the D(2, 1; alpha) superalgebra and their realization on the physical states. The super wave-function encompasses various multiplets of the SU(2)(R) and SU(2)(L) subgroups of D(2, 1; alpha), with fixed isospins. The conformal potential is determined by the external magnetic field in the Wess-Zumino term, whose strength is quantized like in the OSp(4|2) case. As a byproduct, we reveal new invariant subspaces in the enveloping algebra of D(2, 1; alpha) for our quantum realization.