  We discuss integrable extensions of real nonlinear wave equations with
multi-soliton solutions, to their bicomplex, quaternionic, coquaternionic and
octonionic versions. In particular, we investigate these variants for the local
and nonlocal Korteweg-de Vries equation and elaborate on how multi-soliton
solutions with various types of novel qualitative behaviour can be constructed.
Corresponding to the different multicomplex units in these extensions, real,
hyperbolic or imaginary, the wave equations and their solutions exhibit
multiple versions of antilinear or $\mathcal{PT}$-symmetries. Utilizing these
symmetries forces certain components of the conserved quantities to vanish, so
that one may enforce them to be real. We find that symmetrizing the
noncommutative equations is equivalent to imposing a $\mathcal{PT}$-symmetry
for a newly defined imaginary unit from combinations of imaginary and
hyperbolic units in the canonical representation.
