Algebraically closed field s.t. $K\simeq L$, $K\subsetneq L$.

Question

Are there algebraically closed fields $K$ and $L$ that satisfy $K\simeq L$ and $K\subsetneq L$?

Answer 1

There are lots of examples. Any two algebraically closed fields of the same characteristic and the same uncountable cardinality are isomorphic (see here). Hence, for any uncountable algebraically closed field $K$ there is an isomorphism between $K$ and $\overline{K(T)}$, and of course that contains $K$.