Is $[\bar{\mathbb Q}:\bar{\mathbb Q}\cap\mathbb R]=2$? 
Is $[\bar{\mathbb Q}:\bar{\mathbb Q}\cap\mathbb R]=2$ ?

I think it is true that $\bar{\mathbb Q}\cap\mathbb C=\bar{\mathbb Q}$, because I've heard that the closure of the reals is $\mathbb C$. And $\bar{\mathbb Q}$ is a subfield of $\mathbb C$, so using this fact, we have $\bar{\mathbb Q}\cap\mathbb R$, intersection of $2$ fields, again a field.
but does the index necessarily have to be an integer as in the group theory ?

Yes, $[\overline{\mathbb Q}:\overline{\mathbb Q}\cap \mathbb R]=2$, since you attain the former from the latter by adjoining $i$.
The index is always either a positive integer or infinite, since it represents the cardinality of a basis of a vector space. 
