Geometric interpretation of complex numbers and relation with cartesian plane

Question

I was reading about how for the imaginary numbers now called complex numbers Gauss found a geometrical interpretation and were "legalized" in math.
So basically $i$ is a rotation through $90^\circ$ and the complex number $a + bi$ and its operations "move" a point across the cartesian plane.
Assuming my description is correct, it seems to me that essentially the complex numbers are an aggregate of all the pairs of a cartesian plane but what I am not sure if this goes together with the complex number interpretation or not. I think we could also have considered all pair of reals as defining a plane but why was the rotation interpretation required?

Answer 1

"imaginary numbers now called complex numbers" Strictly speaking the imaginary numbers, numbers of the form bi for some real number a, are a subset of the complex numbers, numbers of the form a+ bi for some real numbers a and b.

Yes, the Cartesian coordinate system associates every point in the plane with a pair of numbers, (x, y). The difference is that while the Cartesian coordinate system is purely "geometric", the "complex plane" also has an "algebraic" structure- we can add and multiply complex numbers which we cannot do with points.