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We investigate supergroups with Grassmann parameters replaced by odd Cliffordparameters. The connection with non-anticommutative supersymmetry is discussed.A Berezin-like calculus for odd Clifford variables is introduced. Fermioniccovariant derivatives for supergroups with odd Clifford variables are derived.Applications to supersymmetric quantum mechanics are made. Deformations of theoriginal supersymmetric theories are encountered when the fermionic covariantderivatives do not obey the graded Leibniz property. The simplest non-trivialexample is given by the N=2 SQM with a real $(1,2,1)$ multiplet and a cubicpotential. The action is real. Depending on the overall sign ("Euclidean" or"Lorentzian") of the deformation, a Bender-Boettcher pseudo-hermitianhamiltonian is encountered when solving the equation of motion of the auxiliaryfield. A possible connection of our framework with the Drinfeld twistdeformation of supersymmetry is pointed out.

On Supergroups with Odd Clifford Parameters and Supersymmetry with Modified Leibniz Rule