  A generalized quantization principle is considered, which incorporates
nontrivial commutation relations of the components of the variables of the
quantized theory with the components of the corresponding canonical conjugated
momenta referring to other space-time directions. The corresponding commutation
relations are formulated by using quaternions. At the beginning, this extended
quantization concept is applied to the variables of quantum mechanics. The
resulting Dirac equation and the corresponding generalized expression for plane
waves are formulated and some consequences for quantum field theory are
considered. Later, the quaternionic quantization principle is transferred to
canonical quantum gravity. Within quantum geometrodynamics as well as the
Ashtekar formalism the generalized algebraic properties of the operators
describing the gravitational observables and the corresponding quantum
constraints implied by the generalized representations of these operators are
determined. The generalized algebra also induces commutation relations of the
several components of the quantized variables with each other. Finally, the
quaternionic quantization procedure is also transferred to $\mathcal{N}=1$
supergravity. Accordingly, the quantization principle has to be generalized to
be compatible with Dirac brackets, which appear in canonical quantum
supergravity.
