Why is this definition of complex numbers "informal"?

I'm reading the proofwiki page about complex number: https://proofwiki.org/wiki/Definition:Complex_Number

According to proofwiki there is an informal and formal definitions of complex numbers. The informal definition is that a complex number is equal to $a+bi$ where $a,b \in \Bbb R$ and where $i$ is defined to be the square root of $-1$. What is informal about this definition ?

I guess that this is because you can just define $i$ to be so that $i^2=-1$. But I don't totally get why this is. Would it be formal if I defined $i$ as a number that has all field properties of real numbers, and $i^2=-1$ ?

The only problem I see is that you could argue that there are 2 of those numbers. So which one is meant ? But if you prove that it doesn't matter which one you pick, would this definition then be formal ?

Btw, I'm aware that my reasoning is probably wrong, but I think it is helpfull if I share what is going on in my mind.