Quantum theory (QT) which is one of the basic theories of physics, namely in terms of Schrödinger's 1926 wave functions in general requires the field C of the complex numbers to be formulated. However, even the complex-valued description soon turned out to be insufficient. Incorporating Einstein's theory of Special Relativity (Schrödinger, Klein, Gordon, 1926, Dirac 1928) leads to an equation which requires some coefficients which are hypercomplex. Conventionally the Dirac equation is written using pairwise anti-commuting matrices. However, a unitary ring of square matrices is an - associative - hypercomplex algebra by definition. However, only the algebraic properties of the elements and their relations to one another are important. We hence replace the matrix formulation by a more symbolic one. In the case of the Dirac equation, these elements are called biquaternions. As an algebra over R, the biquaternions are eight-dimensional; as subalgebras, this algebra contains the division ring H of the quaternions at one hand and the algebra C⊗C of the bicomplex numbers at the other, the latter being commutative. As it will later turn out, C⊗C contains pure non-real subalgebras isomorphic to C. Within this paper, we first consider briefly the basics of the non-relativistic and the relativistic quantum theory. Then we introduce general hypercomplex algebras and also show how a relativistic quantum equation like Dirac's one can be formulated using hypercomplex coefficients. Subsequently, some algebraic preconditions for operations within hypercomplex algebras and their subalgebras will be examined. For our purpose equations akin the Schrödinger's one should be able to be set up and solved. Functions of complementary variables like x and p should be Fourier transforms of each other. This should hold within a purely non-real subspace which must hence be a subalgebra. Furthermore, it is an ideal denoted by J. It must be isomorphic to C, hence containing an internal identity element. The bicomplex numbers will turn out to fulfil these preconditions, and therefore, the formalism of QT can be developed within its subalgebras. We also show that bicomplex numbers encourage the definition of several different kinds of conjugates. One of these treats the elements of J precisely as the usual conjugate treats complex numbers. This defines a quantity what we call a modulus which, in contrast to the complex absolute square, remains non-real (but may be called `pseudo-real'). However, we do not conduct an explicit physical interpretation here but we leave this to future examinations.