We study certain new models of supersymmetric quantum mechanics. The explicit form of the corresponding superfield and component actions, as well as of the quantum Hamiltonians and supercharges is given. It is shown that the Hamiltonian H=D*D, where D is flat four-dimensional Dirac operator in an external self-dual gauge background, Abelian or non-Abelian, is supersymmetric with N=4 supersymmetry. A generalization of this Hamiltonian to the motion on a curved conformally flat four-dimensional manifold exists. For an Abelian self-dual background, the corresponding Lagrangian can be derived from certain harmonic superspace expressions. If the Hamiltonian involves a non-Abelian self-dual gauge field, one can construct the Lagrangian formulation of it by introducing auxiliary bosonic variables with Wess-Zumino type action. For a special class of such Lagrangians when the gauge group is SU(2) and the gauge field is expressed in the `t Hooft ansatz form, it is possible to give a superfield description using the harmonic superspace formalism. Independently, a similar system with N=4 supersymmetry in three dimensions also admits the superfield description.