In algebra, one way to understand $\mathbb{C}$ is to consider it as a field extension of $\mathbb{R}$.
What (sometimes) worries me is this: From this point of view, does $\mathbb{C}$ really contain a genuine copy of $\mathbb{R}$? I hope my question makes sense. To me, it is quite obvious that $\mathbb{C}$ contains some field, say $\mathcal{R}$ that is isomorphic to $\mathbb{R}$. But what we really want is that $\mathbb{C}$ contains $\mathbb{R}$ itself?
Update: As far as "algebra" concerns, as long as we are only working with the binary operations $\times$ and $+$, it might be pointless trying to distinguishing the true $\mathbb{R}$ and the isomorphic copy due to isomorphism. However, when we do more complicated things (taking limit, taking integral, considering topological structure, etc.), I assume it would be natural to question whether the same conclusions carries over?
I think this question is going to turn out to be very difficult to answer to your satisfaction - what, after all, is $\mathbb{R}$?
What do I mean? Well, let's say we all agree what the rationals are. Then is a real number
An equivalence class of Cauchy sequences of rationals?
A set of Dedekind cuts in $\mathbb{Q}$?
Note that these are fundamentally different set-theoretic objects: the former is a set of sequences of rationals, while the second is just a set of rationals. And these aren't the only ways to think of $\mathbb{R}$.
So any time you ask a question of the form "Is $\mathbb{R}$ a subset of [thing]?", you have to first specify what exactly you mean by $\mathbb{R}$. And nobody does that . . .
. . . Because what we're really interested in is fields up to isomorphism.
EDIT: This is what I get for writing an answer on a plane! :P
I think my last line - above the edit - may come off as dismissive. It shouldn't. When I write that "what we're really interested in is fields up to isomorphism," I do not mean that questions that dig deeper than isomorphism are non-mathematical. Rather, I mean that when we speak of "the reals" in mathematical practice, this is the viewpoint we're (almost always) tacitly adopting.
What sort of question is "what are the reals exactly?" (Or the rationals. Or the integers.) Well, there's certainly a philosophical aspect, but I'd argue that that doesn't render it non-mathematical. But it does mean we have to be satisfied with answers that are really more about set theory than they are about algebra.
More fundamentally, the real problem is in determining what criteria we use to judge the possible answers. If I give you two definitions of $\mathbb{R}$, how do you tell which one is "more true" (let alone how do we identify one as being exactly right)?
Indeed, one extremely reasonable response is to reject phrases like "the reals" as meaningless - we can talk about complete ordered Archimedean fields, but singling one of them out to be the "right" reals would be bizarre (so the argument goes). In fact, we can view this line of thought as an indictment of set-theoretic foundations in general (look up "structural set theory"). My own philosophical opinions are poorly formed and frequently changing (to make up for the poorly-formedness), but this is a viewpoint I have a lot of sympathy for.