You are very right.

But, might be more complete to say that complex numbers are two-dimensional numbers used to model a lot of different things. So simply saying they are two dimensional numbers only part of the explanation. To fully understand them, you need to consider the notion of a 2-d number fits the application area, and what it means in context.

For example, complex numbers can provide an elegant way to represent rotations, oscillations, waves, etc. making certain calculations much easier.

But complex numbers are also used to extend the real number system in a way that ensures every non-constant polynomial has a root. In that context, they aren't merely a convenient tool for representing two-dimensional phenomena; they help complete the number system itself in a mathematically precise sense.

And then there is quantum mechanics, where complex numbers play a fundamental role in the mathematical description of reality itself.