  We derive and discuss a new type of N=2 supersymmetric quantum mechanical
sigma models which appear when the superfield action of the (1,2,1) multiplets
is modified by adding an imaginary antisymmetric tensor to the target space
metric, thus completing the latter to a non-symmetric Hermitian metric. These
models are not equivalent to the standard de Rham sigma models, but are related
to them through a certain special similarity transformation of the
supercharges. On the other hand, they can be obtained by a Hamiltonian
reduction from the complex supersymmetric N=2 sigma models built on the
multiplets (2,2,0) and describing the Dolbeault complex on the manifolds with
proper isometries. We study in detail the extremal two-dimensional case, when
the target space metric is defined solely by the antisymmetric tensor, and show
that the corresponding quantum systems reveal a hidden N=4 supersymmetry.
