We overcome the barrier of constructing N - 4 superconformal models in one space dimension for more than three particles. The D(2, 1; alpha) superalgebra of our systems is realized on the coordinates and momenta of the particles, their superpartners and one complex pair of harmonic variables. The models are determined by two prepotentials, F and U, which must obey the WDVV and a Killing-type equation plus homogeneity conditions. We investigate permutation-symmetric solutions, with and without translation invariance. Models based on deformed An and BCDn root systems are constructed for any value of alpha, and exceptional F-n-type and super root systems admit solutions as well. Translation-invariant mechanics occurs for any number of particles at alpha=-1/2 (osp(4/2) invariance as a degenerate limit) and for four particles at arbitrary alpha (three series).