Explore the advanced algebraic framework of Dihedron algebra and its application to complex numbers in this 28-minute mathematics lecture. Delve into the properties of 2x2 matrices over general fields, focusing on the crucial conjugation/adjugate operation. Discover how this approach yields both red and green versions of complex numbers, and compare the resulting geometry to that of Quaternions. Learn about quadratic inverse algebras (qia) and their significance in this field. Gain insights into ongoing research and potential future developments in this area of mathematics.