Complex numbers are the key to complex analysis which is different than just real multivariate calculus (it has certain constraints).

> Why don't we just use vectors and make math easier to learn?

We do just use vectors; over complex numbers. You really want both, where the vectors themselves are made up of[1] complex numbers. Using a single complex number to represent a vector has gone out of style in some sense because it's restricted to two dimensions. For example historically it was quite common to represent the velocity of a fluid at each point with a complex number. While you can do this, and there are many of advantages, most don't generalize to higher dimensions.

[1] technically the field is complex numbers https://en.wikipedia.org/wiki/Field_%28mathematics%29

Complicated math concepts (like differential equations) are much easier to learn with a solid background in complex numbers. Vectors are important, but so are complex numbers.

How would you compute exp(v) for a vector v? (for example)

And basically, negative numbers can be avoided as well using the concept of vectors