Adjoin indeterminates to a field making it algebraically closed and locally compact

Question

In order to set up the apparatus for some work I'm doing, I'd like to construct a field with certain properties. I'm hoping this is a standard construction, though I've been unable to find it in the literature, and I hope someone can just point me to a good reference for it.

For the complex numbers $\mathbb{C}$ "all good things are true":

1. a field (closed under the field operations)
2. characteristic 0
3. algebraically closed
4. locally compact under a topology compatible with the field operations (enabling convergence and integration)
5. the finite-dimensional projective spaces $(\mathbb{C}^n \setminus \{0\})/(\mathbb{C}\setminus\{0\}) = \mathbb{C}^n/\mathbb{C}^\times$ are compact. (IIUC, this is a consequence of local compactness.)

I want to adjoin to $\mathbb{C}$ a countable infinity of indeterminates $x_1, x_2, x_3, \ldots$ and retain these properties. Implicitly, the set of indeterminates have a discrete topology; there is no "convergence" of $x_n$ as $n \rightarrow \infty$.

1. Constructing a field is simple; use $\mathbb{C}(x_1, \ldots)$, the field of rational functions over the indeterminates.

2. The characteristic carries over from $\mathbb{C}$.

3. Algebraic closure is obtained by extending the rational functions to Puiseaux series. (Is there a standard symbol for Puiseaux series analogous to $\mathbb{C}((x))$?) IIUC, we want to require that the series are convergent in some ring around 0, so the coefficients are bounded by some exponential function. And not all Puiseaux series of this type will be generated by algebraic closure.

4. IIUC, the Puiseaux series have a natural valucation derived from the natural valuation on $\mathbb{C}$, and the completion of the (convergent) Puiseaux series under that valuation gives the Levi-Civita field.

5. I am hoping this makes the resulting finite-dimensional projective spaces compact so we can integrate on them.

Which means that the extended field also has "all good things are true" in the above sense. I expect that $\mathbb{C}$ could be replaced in this construction by any other field for which "all good things are true".