  A representation of the quadratic Dirac equation and the Maxwell equations in
terms of the three-dimensional universal complex Clifford algebra is given. The
investigation considers a subset of the full algebra, which is isomorphic to
the Baylis algebra. The approach is based on the two Casimir operators of the
Poincare group, the mass operator and the spin operator, which is related to
the Pauli-Lubanski vector. The extension to spherical symmetries is discussed
briefly. The structural difference to the Baylis algebra appears in the shape
of the hyperbolic unit, which plays an integral part in this formalism.
