Example of two non-isomorphic fields which embed inside each other 
Can you find an example of non-isomorphic fields which embed inside each other?

Most probably we can't but I am looking for extraordinary answer...

This is not something I understand well, but I believe you can construct examples as follows.
Let $E$ be an elliptic curve, and let $E'$ be another elliptic curve that is isogenous to $E$, but not isomorphic to $E$.
Then the isogeny should provide an embedding $k(E') \to k(E)$, and its dual should provide an embedding $k(E) \to k(E')$.
