On hypercomplexifying real forms of arbitrary rank

Abstract

For certain problems involving vector fields, it is possible to find an associated imaginary field that, in conjunction with the first, forms a complex field for which the equation can be solved. This result is generalized to arbitrary Clifford algebras, followed by quaternionic vectors as a special case. All results are shown to reduce to the established method of complexifying vector fields. For simplicity, differential forms are used rather than vector notation.