Isomorphism over $mathbb{Q}$ but not isomorphism in $mathbb{C}$

Sorry for my bad English.

Is there example as follows?

$Let$ $K,L$ be subfield of $mathbb{C}$ such that $Lneq K$ as subfield of $mathbb{C}$,
but $Lcong K$ as field (ignoring emmbding to $mathbb{C}$) .