Summary: Using complex notation, we present new simple expressions for two pairs of complex supercharges in HKT (‘hyper-Kähler with torsion’) supersymmetric sigma models. The second pair of supercharges depends on the holomorphic antisymmetric ‘hypercomplex structure’ tensor \(\mathcal{I}_{jk}\) which plays the same role for the HKT models as the complex structure tensor for the Kähler models. When the Hamiltonian and supercharges commute with the momenta conjugate to the imaginary parts of the complex coordinates, one can perform a Hamiltonian reduction. The models thus obtained represent a special class of quasicomplex sigma models introduced recently by E. A. Ivanov and A. V. Smilga [SIGMA, Symmetry Integrability Geom. Methods Appl. 9, Paper 069, 17 p. (2013; Zbl 1287.81108)].