In a discussion regarding the relation between 2D-vectors and complex numbers Jagerber48 states:
This isn't to say there aren't important similarities between the two spaces (they are both 2-dimensional, for example) that are worth keeping in mind to help get intuition to solve various problems.
Does the statement "they are both 2-dimensional" have some mathematical depth or is it only illustrative? The mathematical Wikipedia article on the lookup word 2-dimensional space does not add much more than also point to some kind of similarity between a 2D-plane and the complex plane. If it's not only illustrative; what's required to assert they are both 2-dimensional and hence "similar" in that respect, although different in other aspects?
The complex plane is not just "similar" to a two-dimensional vector space. It is a two-dimensional vector space (over the real numbers). But in addition to this vector space structure it has more parts to it.
Most importantly, there is a certain notion of a product between two complex numbers that yields a new complex number. This structure goes beyond what vector spaces do and it is what makes the complex numbers what they are.
That doesn't stop the complex numbers from being a vector space. And it really shouldn't stop you from visualising the complex numbers in the plane. But it means there is more to them than just addition and length scaling.