I am looking for an example of two fields that are isomorphic, infinite, and not equal.
I have found examples such as $\mathbb{Q}(\sqrt[3]{2}) \cong \mathbb{Q}( \sqrt[3]{2}(\frac{-1 + \sqrt{-3}}{2}) )$ where each of these fields is obtained by adding one of the roots of $x^3 - 2$ to $\mathbb{Q}$. However, I cannot seem to find an example of isomorphic fields that are not obtained in this way, that is, are not obtained by adding different roots of some irreducible polynomial.
Very generally, given any kind of structure, you can get an isomorphic structure that is not equal by just renaming the elements of your structure.
In your case, let $X$ be any infinite field with operations $+_X$ and $\cdot_X$, and let $Y$ be any set which has the same cardinality as $X$ but is not equal to $X$. Let $f:Y\to X$ be some bijection. Define operations of addition and multiplication on $Y$ by $a+_Yb=f^{-1}(f(a)+_Xf(b))$ and $a\cdot_Y b=f^{-1}(f(a)\cdot_X f(b))$. These operations make $Y$ a field, and make $f:Y\to X$ an isomorphism of fields. But $Y$ is not equal to $X$.