There are two perspectives that I have seen presented simultaneously in analysis courses, but which seem to be at odds with each other.
Perspective 1) The real numbers are a subfield of $\mathbb{C}$ because $x = x + 0i$.
Perspective 2) The real numbers are embedded in $\mathbb{C}$ by the mapping $f: \mathbb{R} \to \mathbb{C}, \; x \mapsto x + 0i$.
In this second case, it is not true that $x = x + 0i$, but rather that $f(x) = x + 0i$. Then, it is not that $\mathbb{R}$ is a subfield of $\mathbb{C}$, but rather than the image of $\mathbb{R}$ under the map $f$ is a subfield of $\mathbb{C}$ which obeys precisely the same properties as $\mathbb{R}$ because the field operations in $\mathbb{R}$ are consistent with those in $\mathbb{C}$.
Are these perspectives in fact mutually exclusive? Is this simply an abuse of notation? Is one more typical than the other?
In every construction of the complex field that I am aware of, $\Bbb R\not\subset\Bbb C$. But in each case we identify the real field with a subfield of the complex field. After that, yes, by an abuse of notation, we say that $\Bbb R$ is a subset of $\Bbb C$.