They did, linear algebra and vector calculus are a thing, but complex numbers have certain properties that you don’t get with vectors and that are quite useful and worth studying.

One definition of the complex numbers is the set of tuples (x, y) in R^(2) with the operations of addition: (a,b) + (c,d) = (a+c, b+d) and multiplication: (a,b) * (c,d) = (ac - bd, ad + bc). Then defining i := (0,1) and identifying (x, 0) with the real number x, we can write (a,b) = a + bi.

For various math reasons you only get consistent systems with 2^n dimensions, so after complex you get quaternions with 4, then the next one that works is 8, then 16, etc. They become less useful because you lose various useful features, like you lose commutabiliy with quaternions (eg ab != ba), and every time you double you lose more things.