From my understanding, $C_p$ elements can be written as Hahn–Mal'cev–Neumann series with elements of the form $p^k * u * s$ where $u$ is a unit and $s$ is a root of unity.
Could we as such not work with some cyclotomic extension of the rational numbers, as a sub field of the complex numbers, and simply interpret them (and associated equations) 'p-adically'?
For instance $x^3 + 2x^2 + 4x^1 + 8 = 0$ has roots: $-2$, $2i$, $-2i$ in $C$.
Each has 2-adic valuation equal to one, which fits the Newton polygon of the polynomial, seen 2-adically.
Similarly $x^3 + 2x^2 + 4x^1 - 8 = 0$ has a root: $\frac{-2}{3} + \frac{2 (1 - i \sqrt{3})}{3 (17 + 3 \sqrt{33})^{1/3}} - \frac{1}{3}(1 + i\sqrt{3}) (17 + 3 \sqrt{33})^{1/3}$
Here (using Kronecker-Weber):
$v_2(2(1 - i\sqrt{3})) = 1 + v_2(1 - e^{\frac{2\pi i}{3}} + e^{\frac{4\pi i}{3}}) = 1 + v_2( -2e^{\frac{2\pi i}{3}}) = 2$
and
$17 + 3\sqrt{33} \equiv 20 \pmod{32}$ i.e. $v_2((17 + 3\sqrt{33})^{1/3}) = \frac{2}{3}$
The overall valuation (modulo algebra mistakes) is 1 as expected.
So can we make this kind of inference about polynomials with rational coefficients, over C? That is, that the Newton polygon is descriptive of the roots.