I was thinking about how the imaginary unit $i$ is in fact not defined as "$\sqrt{-1}$" since the square root function is only defined on positive real numbers, but (roughly) as an object such that $i^2=-1$. Then I realised that, in all rigor, it does not make much sense to define $i$ by the value of its square if we haven't defined what it means to square a complex number, i.e. we're defining $i$ according to some operation on it that we haven't defined. In light of that, I reason that if we want to define the set of complex numbers, we must at least say that:
$\mathbb R \subset \mathbb C$
There exists a non-real complex number, namely $i$.
We're defining two operations on the complex numbers, namely $+$ and $\ast$, with such and such properties, in particular the property that $i\ast i=-1$.
In that case, it seems to me that whenever one talks about complex numbers, the notion of the operations defined on it always come with it, especially the notion of complex multiplication. Therefore, does it make sense to talk about the set of complex numbers "on its own", i.e. not $(\mathbb C,+,\ast)$ but just $\mathbb C$, completely independently of any operations defined on its elements ?