This course on complex functions is distinguished by consistently using phase portraits, a special color representation which visualizes functions as images. The text introduces the basic concepts of complex analysis, and, parallel to a systematic investigation of analytic and meromorphic functions, readers learn how properties of a function are reflected in and can be recovered from its phase portrait. The approach emphasizes constructive aspects, it is intuitive, and requires only basic knowledge of calculus. More advanced readers will get a fresh view on classical results and may be inspired by the pictures to new and challenging questions.