Complex-Order and Fractional Derivatives: A First Exploration

This paper presents an independent exploration of fractional and complex-order derivatives, building the framework from algebraic first principles using Gamma function extensions. Without prior exposure to existing fractional calculus literature, I derive formulas for D^α and D^z operators applied to elementary functions, prove fundamental properties (linearity, Index Law, product rules), and explore geometric interpretations through the "D(i) plane." The work concludes with applications to cyclic derivatives and preliminary extensions to matrix orders. Note to Readers: This represents independent rediscovery of classical fractional calculus concepts. I have since learned that this field has extensive existing literature (Riemann-Liouville, Caputo, etc.) and present this work as a pedagogical exercise in mathematical exploration rather than novel research.