The answer is yes, there are such cases. Indeed such cases are in a sense the norm.
If $\kappa$ is an uncountable cardinal, any two algebraically closed fields of cardinality $\kappa$ and characteristic $p$, where $p$ may be $0$, are isomorphic.
In model theory, the standard way to refer to this fact is by saying that the theory of algebraically closed fields of characteristic $p$ is $\kappa$-categorical for any uncountable $\kappa$.
Thus in particular, if $K$ is the field of complex numbers, then the algebraic closure of $K(X)$, where $X$ is transcendental over $K$, is isomorphic to $K$.
The algebraically closed $K$ for which the algebraic closure of $K(X)$ is not isomorphic to $K$ are the "unusual" ones! For example, if $F$ is the field of algebraic numbers, then the only extensions $K$ of $F$ with this property are, up to isomorphism, the algebraic closures of fields obtained by adding a finite number of transcendentals to $F$.