The Clifford algebra Ω generated by the elements {1,ω} with (ω)2 = +1, is an abelian ring of dimension two with properties analogous to the complex field C. The ring Ω has a string of singular inverses, and may be regarded as a ’’singular field’’ which circumvents both the fundamental theorem of algebra and the Frobenius theorem. We construct two associative algebras of dimension four over Ω: the Clifford algebra Ω1and the biquaternions of Clifford Ω2, and demonstrate that both algebras possess inverses everywhere except on a singular region akin to the light cone of the Minkowski space. Matrix representations are discussed, as well as the importance of the algebras Ω1 and Ω2, in the description of physical vector fields.