  We give an algebraic quantization, in the sense of quantum groups, of the
complex Minkowski space, and we examine the real forms corresponding to the
signatures $(3,1)$, $(2,2)$, $(4,0)$, constructing the corresponding quantum
metrics and providing an explicit presentation of the quantized coordinate
algebras. In particular, we focus on the Kleinian signature $(2,2)$. The
quantizations of the complex and real spaces come together with a coaction of
the quantizations of the respective symmetry groups. We also extend such
quantizations to the $\mathcal{N}=1$ supersetting.
