New Complex Numbers

Authors: L. N. Lipatov, M. Rausch de Traubenberg, G. G. Volkov
Title: On the ternary complex analysis and its applications
Link: http://lanl.arxiv.org/abs/0711.0809

Professor Lev Lipatov, together with Rausch de Traubenberg, and Volkov, have accomplished the construction of generalized complex numbers. Instead of the algebraic equation:

x² + y² = z² (Pythagoras Theorem)

they consider:

x₁³ + x₂³ + x₃³ - 3x₁x₂x₃ = z³

With this new structure they solved exactly for the motion of a magnetic monopole, and obtain the transversal cross section of two monopoles colliding.

I can think of at least two other problems to consider with this new structure, hidden variable correlations with three variables in Quantum Mechanics, and three body problems in this same context.