Complex numbers have multiplication defined on them. If you multiply two complex numbers, you get another complex number. (They compose via multiplication in ex

Complex numbers have multiplication defined on them. If you multiply two complex numbers, you get another complex number. (They compose via multiplication in exactly the same way as scaling & rotation operators on 2-dimensional vectors.) If you just have 2-dimensional vectors, there’s no obviously well-defined way to multiply two vectors and get out another vector. In other words, both 2-dimensional vectors and complex numbers are made up of 2 coordinates, but they don’t have the same mathematical structure.