Why does it make sense to talk about the 'set of complex numbers'? In my complex analysis course we've discussed quite a few times the idea that $\mathbb{C}$ is really 'the same thing' as $\mathbb{R}^2$ with the added complex multiplication operation. I've also read a number of the popular posts here including this one: What's the difference between $\mathbb{R}^2$ and the complex plane?.
This post: Is $\mathbb R^2$ a field? explains that the complex numbers can be defined to be the field of $(\mathbb{R}^2,+,*)$, where the operations are the familiar $\mathbb{R}^2$ addition, and complex multiplication.
In my (basic understanding) of  algebra, there is a fundamental difference between the group $(G,*)$, and the set $G$. That is to say that we can meaningfully talk about elements of the set $G$, but not directly about 'elements of the group'. I.e. the group itself is a fundamentally different object to the set $G$, and it tells us about relationships between the elements of $G$.
In this sense, is it possible to talk about 'the set of complex numbers'? If we use the definition of the complex numbers as being the field $(\mathbb{R}^2,+,*)$, then doesn't this really mean that the 'complex numbers' IS a ring? In other words, there is no meaningful way to talk about 'elements' of this ring? If this is the case, then is the set of complex numbers literally $\{(\mathbb{R}^2,+,*)\}$?
The reason I ask is because I am having some conceptual difficulty when confronting the idea of dealing with 'elements' of the complex set. For example if we say $\mathbb{C} = \{x + iy: x,y \in \mathbb{R}\}$, then isn't this just a subset of $\mathbb{R}^2$, since $x + iy = (x,0) + (0,1)*(y,0)$? In this sense this set doesn't actually tell us about the structure imposed on the elements of $\mathbb{R}^2$?
EDIT: I realise my question may be slightly unclear, so I would like to try to express it in the context of the set of natural numbers.
When we talk about $\mathbb{N}$, we are talking about a collection of objects, in which these objects satisfy certain properties, either by definition or by theorem. In particular, when we construct $\mathbb{N}$, each element is precisely defined. So say the symbol $0$ represents the null set, and the symbol $1$ represents $\{0\}$ and so on so forth.
But coming to defining $\mathbb{C}$ now, I am not sure how we do the same construct the set as with $\mathbb{N}$. For example, we want each element of the set of complex numbers to abide by the property of complex multiplication, because this is what makes it fundamentally different from $\mathbb{R}^2$. But this is fundamentally a relationship between two different complex numbers. It requires the operation $*$ to even make sense of. So if we construct a set without the structure, we do literally just end up with the set $\mathbb{C}$ = $\mathbb{R}^2$ because the structure cannot be 'codified' into our construction of the set, because it requires a definition of $*$.
 A: "Element of a group (ring, topological space, etc.)" is simply a common  abreviation for "element of the underlying set of a group (ring, topological space, etc.)".
A: There are algebraic structures like groups, fields, rings, etc., which are tuples of a set and some objects giving the set structure. Usually that's some kind of operation like addition and multiplication, but sometimes special elements of the set are also used. For instance, sometimes groups are defined as a tuple $(G,\ast,e)$ where $G$ is a set, $\ast:G\times G\to G$ a map and $e$ an element of $G$ which together satisfy the group axioms ($\ast$ is associative, for each element of $G$ there exists an inverse, and $e\ast g=g\ast e=g$ for all $g\in G$).
Now when we want to write stuff about the group, writing $(G,\ast,e)$ gets old fast. Which is why essentially everyone just writes down the set instead of the whole tuple, secure in the knowledge that their colleagues will know that they actually mean the tuple when it's clear from context that we're talking about groups. In the same vein, mathematicians will talk about an element of the group when they actually mean an element of the set which is part of the tuple defining the group. But again, all their colleagues are in the know, so it's ok.
The same goes for the complex numbers. Yes, technically, the complex numbers are the tuple $(\mathbb R^2,+,\cdot)$, so an element of "the complex numbers" is not an element of the set $\mathbb C:=\mathbb R^2$. But we talk about elements of the complex numbers anyway because it would be tiresome to talk about elements of the underlying set of the complex numbers. Everyone knows what you mean anyway.
TL;DR: Mathematicians are lazy, so they talk about elements of a group even if it's not technically correct.
A: In any case, there is a trivial isomorphism between $(\mathbb R^2,+,\times)$ where the multiplication is that of the complex, and $(\mathbb C,+,\times)$. From an abstract point of view, these structures are interchangeable.
I don't see any reason to avoid the definition of a set denoted as $\mathbb C$, where the elements are equivalently written like
$$z:=(a,b)$$ or $$z:=a+ib$$ where $a,b\in\mathbb R$. Whether it is considered a different set from $\mathbb R^2$ or not seems a useless/irrelevant question. Anyway, if they are considered different, allowing to mix elements of $\mathbb C$ and $\mathbb R^2$ (e.g. defining addition between them) would seem a paranoid idea.
A: There are good answers to the question, but having thought more about Robert Israel's comment I believe I have gained some understanding which may be useful for others.
The problem attempts to 'capture', in some sense, objects and put them into a set that I would call $\mathbb{C}$. This is much like being able to point to $5$ and saying the object that symbol represents is in $\mathbb{N}$. There is an assumption that an element of $\mathbb{R}^2$ should not be the same as an element of $\mathbb{C}$ (because we all know complex numbers are not the same as ordered pairs of real numbers), but this is erroneous .
Specifically, we call elements of $\mathbb{R}^2$ 'complex numbers', when the context of dealing with these elements involves the complex multiplication operation.
So it makes no sense to look for fundamentally different mathematical objects from those in $\mathbb{R}^2$ to put in $\mathbb{C}$, because $\mathbb{C}$ IS $\mathbb{R}^2$. The difference between the two is not due to a difference in elements of a set, it is due to defining the relationship between the elements of $\mathbb{R}^2$, the complex multiplication. I.e., the sets are the same but in one context there is structure on that set, in the other context there isn't.
This question is a bit like asking what are the elements that make up the group $(G,*)$. They're just those elements of $G$ (as the comment notes), but $(G,*)$ gives us important new information on the relationship between those elements of $G$, and that extra information doesn't come from 'changing' the elements of $G$ in any way.
