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 $*$.
Best Answer
"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.)".