Is there a general notion/description of fields $K$ such that the image of any embedding $K \hookrightarrow \mathbb{C}$ is invariant under complex conjugation, thus inducing an involution on $K$ which is independent on the chosen embedding into $\mathbb{C}$ but gives complex conjugation when choosing one?

Examples of such fields are $\mathbb{Q}$, CM-fields like cyclotomic fields, but also $\mathbb{R}$ (which is not a number field, I want to include this case!).

In the book "Unitary Reflection Groups" by Lehrer-Taylor on page 20 (see http://books.google.com/books?id=7QSFEnNh7WkC&lpg=PP1&pg=PA20#v=onepage&q&f=false) it is somehow mentioned that any abelian number field has this property. Is this correct? I doubt this but would be happy if so.

I would like to use this to define the notion of inner products for vector spaces for fields different from $\mathbb{C}$ and $\mathbb{R}$. This is also the context in which Lehrer-Taylor use this. I think it's not a good idea to just take a subfield of $\mathbb{C}$ which is invariant under complex conjugation since then all notions depend on the chosen embedding. Any ideas on how to properly do this are welcome, too.