Conjugation and Moduli in Tricomplex Numbers

This paper develops a systematic framework for the conjugation theory and associated moduli in the tricomplex algebra C_3. The presence of three mutually commuting imaginary units i_1,i_2,i_3 gives rise to a rich family of involutive symmetries extending the classical complex conjugation. We construct and characterize the three fundamental conjugations that individually reverse the signs of i_1,i_2,i_3 and derive four additional composite conjugations generated by their compositions. The resulting seven conjugation operators form an Abelian group under composition and determine a hierarchy of algebraic invariants within C_3. Using these operators, several generalized moduli are introduced and analyzed, including their behavior on bicomplex and tricomplex subalgebras and their multiplicative properties, which extend the classical notion of modulus to higher-dimensional multicomplex structures. The action of conjugation on polynomial equations is also examined, leading to symmetry relations for roots and orbit-based factorization patterns. These results provide a unified extension of classical norm and conjugation theory to higher multicomplex systems and establish foundational tools for the analytic study of tricomplex-valued functions.