Donets, et al., wrote about such stuff back in 1999. I summarize:

The multidimensional N=4 supersymmetric quantum mechanics (SUSY QM) is constructed using the superfield approach. As a result, the component form of the classical and quantum Lagrangian and Hamiltonian is obtained. In the considered SUSY QM both classical and quantum N=4 algebras include central charges, and this opens various possibilities for partial supersymmetry breaking. It is shown that quantum mechanical models with one quarter, one half and three quarters of unbroken(broken) supersymmetries can exist in the framework of the multidimensional N=4 SUSY QM, while the one-dimensional N=4 SUSY QM, constructed earlier, admits only one half or total supersymmetry breakdown. We illustrate the constructed general formalism, as well as all possible cases of the partial SUSY breaking taking as an example a direct multidimensional generalization of the one-dimensional N=4 superconformal quantum mechanical model. Some open questions and possible applications of the constructed multidimensional N=4 SUSY QM to the known exactly integrable systems and problems of quantum cosmology are briefly discussed.

I also recommend Chia-Chun Chou and Donald Kouri's article Multidimensional Supersymmetric Quantum Mechanics: A Scalar Hamiltonian Approach to Excited States by the Imaginary Time Propagation Method (J. Phys. Chem. A, 2013, 117 (16), pp 3449–3457). If anybody's excited it's Gluon the Duckdog!