Here is a different way to think about how these fields arise.

In characteristic zero, all such fields in any uncountable cardinality $\kappa$ can be viewed as arising as a nonstandard version of the algebraic closure the rationals.

That is, take any nonstandard model of arithmetic $M$ of uncountable size $\kappa$, or an $\omega$-nonstandard model of set theory with $\kappa$ many natural numbers, if you like. Let $\mathbb{Q}^*$ be the nonstandard version of the rational field as $M$ sees it, and let $\overline{\mathbb{Q}^*}$ be the algebraic closure of $\mathbb{Q}^*$ as it is constructed in $M$. This is an algebraically closed field of characteristic zero and of the desired cardinality $\kappa$.

In particular, even the complex field $\mathbb{C}$ itself can be seen as the algebraic closure of the rationals $\mathbb{Q}^*$ of a nonstandard model of arithmetic or set theory, and $\mathbb{Q}^*$ is the quotient field of its integer part $\mathbb{Z}^*$.

In this sense, every uncountable algebraically closed field of characteristic zero is exactly a nonstandard version of the algebraic closure of $\mathbb{Q}$.

Note that the algebraic closure of a field as computed inside $M$ is more generous than the actual algebraic closure, since a nonstandard model will be wanting to include all the roots of the various nonstandard polynomials, as well as the standard ones.

I made a blog post, Nonstandard models of arithmetic arise in the complex numbers, when this phenomenon arose a few weeks ago at our seminar.

Here is a different way to think about how these fields arise.

In characteristic zero, all such fields in any uncountable cardinality $\kappa$ can be viewed as arising as a nonstandard version of the algebraic closure the rationals.

That is, take any nonstandard model of arithmetic $M$ of uncountable size $\kappa$, or an $\omega$-nonstandard model of set theory with $\kappa$ many natural numbers, if you like. Let $\mathbb{Q}^*$ be the nonstandard version of the rational field as $M$ sees it, and let $\overline{\mathbb{Q}^*}$ be the algebraic closure of $\mathbb{Q}^*$ as it is constructed in $M$. This is an algebraically closed field of characteristic zero and of the desired cardinality $\kappa$.

In particular, even the complex field $\mathbb{C}$ itself can be seen as the algebraic closure of the rationals $\mathbb{Q}^*$ taken inside a nonstandard model of arithmetic or set theory, and $\mathbb{Q}^*$ is the quotient field there of its integer part $\mathbb{Z}^*$.

In this sense, every uncountable algebraically closed field of characteristic zero is exactly a nonstandard version of the algebraic closure of $\mathbb{Q}$.

Note that the algebraic closure of a field as computed inside $M$ is more generous than the actual algebraic closure, since a nonstandard model will be wanting to include all the roots of the various nonstandard polynomials, as well as the standard ones.

I made a blog post, Nonstandard models of arithmetic arise in the complex numbers, when this phenomenon arose a few weeks ago at our seminar.

Here is a different way to think about how these fields arise.

In characteristic zero, all such fields in any uncountable cardinality $\kappa$ can be viewed as arising as a nonstandard version of the algebraic closure the rationals.

That is, take any nonstandard model of arithmetic $M$ of uncountable size $\kappa$, or an $\omega$-nonstandard model of set theory with $\kappa$ many natural numbers, if you like. Let $\mathbb{Q}^*$ be the nonstandard version of the rational field as $M$ sees it, and let $\overline{\mathbb{Q}^*}$ be the algebraic closure of $\mathbb{Q}^*$ as it is constructed in $M$. This is an algebraically closed field of characteristic zero and of the desired cardinality $\kappa$.

In particular, even the complex field $\mathbb{C}$ itself can be seen as the algebraic closure of the rationals $\mathbb{Q}^*$ of a nonstandard model of arithmetic or set theory. Thus, $\mathbb{C}$ is a degree-two extension of a nonarchimedean ordered field $\mathbb{Q}^*$, which is the quotient field of its integer part $\mathbb{Z}^*$.

In this sense, every uncountable algebraically closed field of characteristic zero is exactly a nonstandard version of the algebraic closure of $\mathbb{Q}$.

Note that the algebraic closure of a field as computed inside $M$ is more generous than the actual algebraic closure, since a nonstandard model will be wanting to include all the roots of the various nonstandard polynomials, as well as the standard ones.

I made a blog post, Nonstandard models of arithmetic arise in the complex numbers, when this phenomenon arose a few weeks ago at our seminar.

Here is a different way to think about how these fields arise.

In characteristic zero, all such fields in any uncountable cardinality $\kappa$ can be viewed as arising as a nonstandard version of the algebraic closure the rationals.

That is, take any nonstandard model of arithmetic $M$ of uncountable size $\kappa$, or an $\omega$-nonstandard model of set theory with $\kappa$ many natural numbers, if you like. Let $\mathbb{Q}^*$ be the nonstandard version of the rational field as $M$ sees it, and let $\overline{\mathbb{Q}^*}$ be the algebraic closure of $\mathbb{Q}^*$ as it is constructed in $M$. This is an algebraically closed field of characteristic zero and of the desired cardinality $\kappa$.

In particular, even the complex field $\mathbb{C}$ itself can be seen as the algebraic closure of the rationals $\mathbb{Q}^*$ of a nonstandard model of arithmetic or set theory, and $\mathbb{Q}^*$ is the quotient field of its integer part $\mathbb{Z}^*$.

In this sense, every uncountable algebraically closed field of characteristic zero is exactly a nonstandard version of the algebraic closure of $\mathbb{Q}$.

Note that the algebraic closure of a field as computed inside $M$ is more generous than the actual algebraic closure, since a nonstandard model will be wanting to include all the roots of the various nonstandard polynomials, as well as the standard ones.

I made a blog post, Nonstandard models of arithmetic arise in the complex numbers, when this phenomenon arose a few weeks ago at our seminar.