  We consider a deformation of N=1 four dimensional Minkowski superspace where
odd coordinates $\theta^{\alpha}$ do not anticommute. We define supersymmetric
and associative star product and show how the remaining (anti)commutation
relations among the superspace coordinates are modified. In particular, the
even coordinates do not commute as well. We also study chiral and vector
superfields and their interactions. Suprisingly we find that ordinary
undeformed N=1 supersymmetric field theories are compatible with the deformed
supersymmetry considered.
