  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. As a new explicit example, the N=4 mechanics with Yang
monopole in R^5 (= instanton on S^4) is considered.
  Independently, a similar system with N=4 supersymmetry in three dimensions
also admits the superfield description. Although the three-dimensional system
involves different superfields, its component Lagrangian and Hamiltonian appear
to be the three-dimensional reduction of the mentioned four-dimensional system.
The off-shell N=4 supersymmetry requires the gauge field to be a static form of
the 't Hooft ansatz for the four-dimensional self-dual SU(2) gauge fields, that
is a particular solution of Bogomolny equations for BPS monopoles.
