As a purely algebraic alternative to model theoretic approaches to nonstandard algebraic closure in characteristic $0$, we can also constructively build them in the regular old universe out of the ordinals.

Let $\alpha>0$ be an ordinal, and consider the field of fractions $Frac(\mathfrak{G}(\omega^{\omega^\alpha}))=\mathbb{Q}^{\omega^{\omega^\alpha}}$ of the Grothendieck ring $\mathfrak{G}(\omega^{\omega^\alpha})=\mathbb{Z}^{\omega^{\omega^\alpha}}$ of the $\delta$-number $\omega^{\omega^\alpha}$ under natural (Hessenberg) operations. This is essentially a nonstandard model of the rationals* where all the ordinals up to $\omega^{\omega^\alpha}$ are now treated as the 'natural numbers' and $\mathbb{Z}^{\omega^{\omega^\alpha}}$ is treated as the 'integer subring' of our field.

We can then move into an algebraic closure $\overline{\mathbb{Q}}^{\omega^{\omega^\alpha}}$, where we allow 'polynomials' to really be elements of the monoid algebra $\mathbb{Q}^{\omega^{\omega^\omega}}(X^{\omega^{\omega^\alpha}})$ -- these are expressions of the form $$\sum_{i<n}q_iX^{\alpha_i}$$ with $q_i\in\mathbb{Q}^{\omega^{\omega^\omega}}$ and $\alpha_i<\omega^{\omega^\alpha}$ for all $i<n$. This gives us '$\beta^{th}$ roots' for members of $\mathbb{Q}^{\omega^{\omega^\alpha}}$ for all $\beta<\omega^{\omega^\alpha}$, as opposed to only finite roots which are provided by algebraic closure using standard polynomials.

There is some subtlety here when trying to approach this without model theory guaranteeing the existence of a way to evaluate $q_iX^{\alpha_i}$ as a function on $\mathbb{Q}^{\omega^{\omega^\alpha}}$ when $\omega\leq\alpha_i$ -- we can get around this by embedding $\mathbb{Q}^{\omega^{\omega^\alpha}}$ in the Surreals $N_0$ and looking at what the Surreal exponential function does to its image.**

You can also do modular arithmetic in $\omega^{\omega^\alpha}$ for ordinals $\beta\geq\omega$ and proceed to produce a nonstandard $\mathbb{Q}^{\omega^{\omega^\alpha}}_p$ for $p\geq\omega$ prime, and then algebraically close that too. We could also just use all of the ordinals instead of stopping at $\omega^{\omega^\alpha}$, in which case we produce a proper class sized version of the above.

I'm currently researching these notions to better understand the Surreals, but they seem to provide a constructive model of many nonstandard constructions usually accessed through model theoretic techniques.

*I am not sure if this actually qualifies as a nonstandard model of the rationals -- I would appreciate it if someone with more expertise than I could verify/refute this.

** I am currently checking if $\mathbb{Q}^{\omega^{\omega^\alpha}}$ embedded in $N_0$ 'preserves exponentiation' to well-define it on the preimage; I believe it should, but this is just intuition for now.

As a purely algebraic alternative to model theoretic approaches to nonstandard algebraic closure in characteristic $0$, we can also constructively build them in the regular old universe out of the ordinals.

Let $\alpha>0$ be an ordinal, and consider the field of fractions $Frac(\mathfrak{G}(\omega^{\omega^\alpha}))=\mathbb{Q}^{\omega^{\omega^\alpha}}$ of the Grothendieck ring $\mathfrak{G}(\omega^{\omega^\alpha})=\mathbb{Z}^{\omega^{\omega^\alpha}}$ of the $\delta$-number $\omega^{\omega^\alpha}$ under natural (Hessenberg) operations. This is a non Archimedean ordered field (thusly it properly contains the rationals) with no nontrivial roots where all the ordinals up to $\omega^{\omega^\alpha}$ are now treated as the 'natural numbers' and $\mathbb{Z}^{\omega^{\omega^\alpha}}$ is treated as the 'integer subring' of our field.

We can then move into an algebraic closure $\overline{\mathbb{Q}}^{\omega^{\omega^\alpha}}$, where we allow 'polynomials' to really be elements of the monoid algebra $\mathbb{Q}^{\omega^{\omega^\omega}}(X^{\omega^{\omega^\alpha}})$ -- these are expressions of the form $$\sum_{i<n}q_iX^{\alpha_i}$$ with $q_i\in\mathbb{Q}^{\omega^{\omega^\omega}}$ and $\alpha_i<\omega^{\omega^\alpha}$ for all $i<n$. This gives us '$\beta^{th}$ roots' for members of $\mathbb{Q}^{\omega^{\omega^\alpha}}$ for all $\beta<\omega^{\omega^\alpha}$, as opposed to only finite roots which are provided by algebraic closure using standard polynomials.

There is some subtlety here when trying to approach this without model theory guaranteeing the existence of a way to evaluate $q_iX^{\alpha_i}$ as a function on $\mathbb{Q}^{\omega^{\omega^\alpha}}$ when $\omega\leq\alpha_i$ -- we can get around this by embedding $\mathbb{Q}^{\omega^{\omega^\alpha}}$ in the Surreals $N_0$ and looking at what the Surreal exponential function does to its image.**

You can also do modular arithmetic in $\omega^{\omega^\alpha}$ for ordinals $\beta\geq\omega$ and proceed to produce a nonstandard $\mathbb{Q}^{\omega^{\omega^\alpha}}_p$ for $p\geq\omega$ prime, and then algebraically close that too. We could also just use all of the ordinals instead of stopping at $\omega^{\omega^\alpha}$, in which case we produce a proper class sized version of the above.

I'm currently researching these notions to better understand the Surreals, but they seem to provide a constructive model of many nonstandard constructions usually accessed through model theoretic techniques.

*This used to claim that $\mathbb{Q}^{\omega^{\omega^\alpha}}$ was essentially a nonstandard model of the rationals, which is incorrect (thanks to nombre on this).

** I am currently checking if $\mathbb{Q}^{\omega^{\omega^\alpha}}$ embedded in $N_0$ 'preserves exponentiation' to well-define it on the preimage; I believe it should, but this is just intuition for now.