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The mathematical concept of bicomplex numbers (quaternions) is introduced in electromagnetics, and is directly applied to the derivation of analytical solutions of Maxwell's equations. It is demonstrated that, with the assistance of a bicomplex vector field, a novel entity combining both the electric and the magnetic fields, the number of unknown quantities is practically reduced by half, whereas the Helmholtz equation is no longer necessary in the development of the final solution. The most important advantage of the technique is revealed in the analysis of electromagnetic propagation through inhomogeneous media, where the coefficients of the (second order) Helmholtz equation are variable, causing severe complications to the solution procedure. Unlike conventional methods, bicomplex algebra invokes merely first order differential equations, solvable even when their coefficients vary, and hence enables the extraction of several closed form solutions, not easily derivable via standard analytical techniques.