Let $V[G]$ be a generic extension of $V$ by adding a new Cohen real (or generally a generic extension which adds new reals and do not blow up the power of the continuum). Working in $V[G],$ we can consider the following structures:

1) $(\mathbb{C}^{V[G]}, +, ., 0, 1)$; the field of complex numbers as computed in $V[G].$

2) $(\mathbb{C}^{V}, +, ., 0, 1)$; the field of complex numbers as computed in $V.$

Both of these structures are models of algebraically closed fields of characteristic zero and are of the size of the continuum, so it follows that

$(\mathbb{C}^{V[G]}, +, ., 0, 1) \cong (\mathbb{C}^{V}, +, ., 0, 1)$.

**Question.** Is it possible to define an explicit isomorphism between the above structures? or is it possible to show that there is no definable isomorphism?