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The main idea of the paper is that fast algorithms, like FFT, can be made more efficient in the context of an algebra, rather than in the more singular quaternion or complex algebras structure. However, the complex algebra structure can then be recovered as a projection from the larger algebra in which it is embedded. Namely, the 12-dimensional algebra (hurwitzion algebra) having the basis elements associated with the integer Hurwitz quaternions is introduced. The computational aspects of the hurwitzion arithmetic are considered. The overlapped fast algorithms of two-dimensional discrete Fourier transform of an RGB image are also developed. [ABSTRACT FROM AUTHOR]