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Quantum theory (QT), namely in terms of Schr\"odinger's 1926 wave functionsin general requires complex numbers to be formulated. However, it soon turnedout to even require some hypercomplex algebra. Incorporating Special Relativityleads to an equation (Dirac 1928) requiring pairwise anti-commutingcoefficients, usually $4\times 4$ matrices. A unitary ring of square matricesis an associative hypercomplex algebra by definition. Since only the algebraicproperties and relations of the elements matter, we replace the matrices bybiquaternions. In this paper, we first consider the basics of non-relativisticand relativistic QT. Then we introduce general hypercomplex algebras and alsoshow how a relativistic quantum equation like Dirac's one can be formulatedusing biquaternions. Subsequently, some algebraic preconditions for operationswithin hypercomplex algebras and their subalgebras will be examined. For ourpurpose equations akin to Schr\"odinger's should be able to be set up andsolved. Functions of complementary variables should be Fourier transforms ofeach other. This should hold within a purely non-real subspace which must hencebe a subalgebra. Furthermore, it is an ideal denoted by $\mathcal{J}$. It mustbe isomorphic to $\mathbb{C}$, hence containing an internal identity element.The bicomplex numbers will turn out to fulfil these preconditions, andtherefore, the formalism of QT can be developed within its subalgebras. We alsoshow that bicomplex numbers encourage the definition of several different kindsof conjugates. One of these treats the elements of $\mathcal{J}$ like the usualconjugate treats complex numbers. This defines a quantity what we call amodulus which, in contrast to the complex absolute square, remains non-real(but may be called `pseudo-real'). However, we do not conduct an explicitphysical interpretation here but we leave this to future examinations.

Hypercomplex Algebras and their application to the mathematical formulation of Quantum Theory