In courses on Linear Algebra and Algebra, students are exposed to the idea that algebra over complex numbers is much nicer. For example, linear maps between complex vector spaces are always diagonalisable. These nice properties are a consequence of the fact that complex numbers form an algebraically closed field. But, proof that the field of complex numbers is algebraically closed — an algebraic property — uses the analysis in an essential way. Moreover, the standard functions (polynomial, exponential, logarithmic, and trigonometric) will be revealed to have much more in common. Taylor series expansions of functions and their regions of convergence become far less mysterious when viewed in the complex plane. Fundamental ideas and methods from multivariable calculus often make more sense when viewed through the lens of complex function integration theory.
Complex analysis also forms a launch pad to further studies in both pure and applied
mathematics. It forms the foundation of various areas like dynamics, number theory,
differential geometry, various transforms (Fourier, Laplace, etc), differential equations
etc. Thus, the course is ideal for all students who want to do higher studies in pure
mathematics, applied mathematics, and physics.