Does symbol "$+$" denote an operation in the notation of a complex number: "$a+ib$"? In case it does, which operation does "$+$" denote?

There is indeed a very annoying abuse of notation here. The short version is that the "$+$" in "$a+bi$" - in the context of defining the complex numbers - is being used as a purely formal symbol; that said, after having made sense of the complex numbers it can be conflated with complex addition.

An actually formal way to construct $\mathbb{C}$ from $\mathbb{R}$ is the following:

• A complex number is an ordered pair $(a,b)$ with $a,b\in\mathbb{R}$.

• We define complex addition and complex multiplication by $$(a,b)+_\mathbb{C}(c,d)=(a+c,b+d)$$ and $$(a,b)\times_\mathbb{C}(c,d)=(a\times c-b\times d, a\times d+b\times c)$$ respectively. Note that we're using the symbols "$+$," "$-$," and "$\times$" here in the context of real numbers - we're assuming those have already been defined (we're building $\mathbb{C}$ from $\mathbb{R}$).

• We then introduce some shorthand: for real numbers $a$ and $b$, the expression "$a+bi$" is used to denote $(a,b)$, "$a$" is shorthand for $(a,0)$, and "$bi$" is shorthand for $(0,b)$. We then note that "$a+bi=a+bi$," in the sense that $$a+bi=(a,b)=(a,0)+_\mathbb{C}(0,b)=a+_\mathbb{C}bi$$ (cringing a bit as we do so).

Basically, what's happening in the usual construction of the complex numbers is that we're overloading the symbol "$+$" horribly; this can in fact be untangled, but you're absolutely right to view it with skepticism (and it's bad practice in general to construct a new object so cavalierly).

This old answer of mine explains how properties of $\mathbb{C}$ can be rigorously proved from such a rigorous construction, and may help clarify things. Additionally, it's worth noting that this sort of notational mess isn't unique to the complex numbers - the same issue can crop up with the construction of even very simple field extensions (see this old answer of mine).

Some would say: we identify a real number $a$ with the complex number $(a,0)$. Then, using this identification, $$ (a,b) = (a,0)+(0,b) = (a,0)+(0,1)(b,0)= a+ib . $$ If we say it that way, then the "$+$" is complex addition. And (with this identificaion) every real number is also a complex number.

Maybe a teacher would (to start with) use a different notation for the real number $a$ and the complex number $a$. But after a while that different notation would be dropped, and the "identification" would be understood.

We have similar things at more elementary level. A natural number is "identified" with an integer. An integer is "identified" with a rational number. A rational number is "identified" with a real number. Should we, in fact, keep different notations for all these?