On the complexifications of the Euclidean $R^n$ spaces and the n-dimensional generalization of Pithagore theorem

We will discuss the following results C_n complexification of R(n) spaces, C_n structure and the invariant surfaces C_n holomorphicity and harmonicity. We also consider the link between C_n holomorphicity and the origin of spin 1/n. In our approach appears a new geometry and N-ary algebras/symmetries.