For what fields $K$ do there exist finite extensions $E/K$ (of degree $>1$) such that $K\cong E$?
If $K$ has finite degree over its prime field, then any finite extension $E/K$ has greater degree over the prime field, therefore $E\not\cong K$.
On the other hand, if $K=k(x)$ for some field $k$, then adjoining a formal square root $y$ of $x$ gives an extension $K(y)/K$ of degree $2$ such that $K\cong K(y)$, since $K=k(x)\cong k(y)=K(y)$.
I suspect the same holds for any field which contains an element which is transcendental over its prime subfield. However, I'm at a loss for how to deal with cases like infinite algebraic extensions.