We extend the multi-valued radical representation previously developed for real Brioschi quintics to the fully complex case. For any complex parameter C C, the Brioschi normal form y^5 - 5y^3 + 5y = C is solved by the same radical expression: y = 5C{2 + (C{2) ^2 - 1}} + 5C{2 - (C{2) ^2 - 1}}. However, the branch cuts of the square root and the fifth root now require a thorough analysis in the complex plane. We identify branch points at C = 2 and at infinity, define principal branches consistent with the real case, and provide a complete algorithm to generate all five roots for any complex C. We also study the Riemann surface of the function y (C) and its monodromy around the branch points. Numerical examples with complex coefficients are given, including full tables of five roots for various complex C. Classical reductions (Tschirnhaus, Bring-Jerrard, Brioschi-Klein) are extended to complex coefficients, enabling the solution of any complex quintic. This work complements our previous real-case paper and opens the door to applications in dissipative systems, PT-symmetric quantum mechanics, and complex analysis. Keywords: Complex quintic equations, Brioschi normal form, multi-valued radical representation, branch cut, Riemann surface, complex Tschirnhaus transformation, monodromy, PT-symmetry.
Waleed mohamed khalaf Moqadem (Fri,) studied this question.