In the following I will describe a proposal for the p-adic expansion of the elements of the algebraic closure $\overline{\mathbb{Q}_p}$ of $\mathbb{Q}_p$. My question is if this "conjecture" has been proved or disproved before.

Consider the following result proved in "Local Fields" Berlin. (1980) by Serre, J. P.:

Theorem:If $R$ is a perfect field of characteristic $p>0$, then there exists a unique ring $W(R)$ with characteristic $0$ with discrete valuation $v$ such that the residue field is $R$, $v(p)=1\in\mathbb{Z}$, and $W(R)$ is complete with respect to $v$. Also, the field of fractions $F(W(R))$ is the only field with characteristic $0$ with discrete valuation $v$ such that the residue field is $R$, $v(p)=1\in\mathbb{Z}$, and $F(W(R))$ is complete with respect to $v$.

In this context, the valuation ring of $F(W(R))$ is $W(R)$ the ring of Witt vectors with coefficients in $R$. For example, if $R=\mathbb{F}_p$, then $W(R)=\mathbb{Z}_p$ and $F(W(R))=\mathbb{Q}_p$.

In the article "Maximally complete fields." Enseign. Math.(2) 39.1-2 (1993): 87-106, @Bjorn Poonen describes the unique maximally complete immediate extension field of $\overline{\mathbb{Q}_p}$ as follows:

let $W(\overline{\mathbb{F}_p})((t^\mathbb{Q}))$ be the ring of all series $\sum_{g}\alpha_gt^g$ where $g\in\mathbb{Q}$, $\alpha_g\in W(\overline{\mathbb{F}_p})$, and $\{g:\alpha_g\neq0\}$ is well-ordered.
A series $\sum_{g}\alpha_gt^g\in W(\overline{\mathbb{F}_p})((t^\mathbb{Q}))$
is *null* if for all $g\in\mathbb{Q}$, $\sum_{n\in\mathbb{Z}}\alpha_{g+n}p^n=0$ in $F(W(\overline{\mathbb{F}_p})).$
The set $N$ of all null series is an ideal of $W(\overline{\mathbb{F}_p})((t^\mathbb{Q}))$ that contains the polynomial $t-p$. The quotient $$L=W(\overline{\mathbb{F}_p})((t^\mathbb{Q}))/N$$ is a field of characteristic $0$, valuation ring $\mathbb{Q}$ and residue class field $\overline{\mathbb{F}_p}$.
Also $L$ is algebraically closed, and it is the maximally complete immediate extension of $\overline{\mathbb{Q}_p}$. Furthermore, if $S\subset W(\overline{\mathbb{F}_p})$ is a set of representatives of the classes of $\overline{\mathbb{F}_p}$, then each series $\sum_{g}\alpha_gt^g$ in $W(\overline{\mathbb{F}_p})((t^\mathbb{Q}))$ is null equivalent to a unique series of the form $\sum_{g}\beta_gt^g$ for $\beta_g\in S$. In other words, we have the following expansion for the elements of $L$:

$$L=\bigg\{\sum_{g\in\mathbb{Q}}\beta_gp^g:\beta_g\in S, \{g:\beta_g\neq0\}\mbox{ is well-ordered}\bigg\}.$$

Notice the analogy with the $p$-adic expansion for elements of $\mathbb{Q}_p$: $$\mathbb{Q}_p=\mathbb{Z}_p((t^\mathbb{Z}))/N=W(\mathbb{F}_p)((t^\mathbb{Z}))/N=\bigg\{\sum_{n=m}^\infty\beta_np^n:m\in\mathbb{Z},\beta_n=0,1,\dots,p-1 \bigg\}.$$

Now let's consider the following chain of valued field extensions: $$\mathbb{Q}_p((x))\subset \overline{\mathbb{Q}_p}((x))\subset \overline{\overline{\mathbb{Q}_p}((x))}= \bigcup_{n=1}^\infty \overline{\mathbb{Q}_p}((x^\frac{1}{n}))\subset\overline{\mathbb{Q}_p}((t^\mathbb{Q}))$$

Also consider the following chain of rings: $$\mathbb{Z}_p((x))\subset W(\overline{\mathbb{F}_p})((x))\subset \bigcup_{n=1}^\infty W(\overline{\mathbb{F}_p})((x^\frac{1}{n}))\subset W(\overline{\mathbb{F}_p})((t^\mathbb{Q}))$$

By taking the quotient of these rings by their ideals of null series, we obtain the following chain of field extensions: $$\mathbb{Q}_p= \mathbb{Z}_p((x))/N\subset W(\overline{\mathbb{F}_p})((x))/N\subset \bigcup_{n=1}^\infty W(\overline{\mathbb{F}_p})((x^\frac{1}{n}))/N\subset W(\overline{\mathbb{F}_p})((t^\mathbb{Q}))/N=L$$ where there is an abuse of notation for the ideals $N$ (The N's of different quotients are different).

Conjecture 1:The relation

$$\overline{\overline{\mathbb{Q}_p}((x))}= \bigcup_{n=1}^\infty \overline{\mathbb{Q}_p}((x^\frac{1}{n}))$$ implies the relation $$\overline{\mathbb{Q}_p}= \bigcup_{n=1}^\infty W(\overline{\mathbb{F}_p})((x^\frac{1}{n}))/N= \bigg\{\sum_{n=m}^\infty\beta_np^\frac{n}{N}:m,N\in\mathbb{Z},N\geq1,\beta_n\in S \bigg\},$$ where $S\subset W(\overline{\mathbb{F}_p})$ is a set of representatives of the classes of $\overline{\mathbb{F}_p}$.

**Edit:** As it is noticed in the comments below, the conjecture 1 is false, since $\sqrt{-1}\in\overline{\mathbb{Q}_2}$ cannot be represented in the proposed form. Also notice that $W(\overline{\mathbb{F}_p})\neq\{x\in\overline{\mathbb{Q}_p}:|x|_p\leq1\}$ since the former has a discrete valuation while latter has a dense valuation. Also I just noticed the following result, which follows immediately by the theorem stated above and the properties of the maximally complete field $L$ described by Bjorn Poonen in its paper also mentioned above. The notations are defined above.

Proposition:If $R$ is a perfect field of characteristic $p>0$, then $W(R)((x))/N$ is the only field of characteristic $0$, with discrete valuation $v$ such that the residue class field is $R$, $v(p)=1\in\mathbb{Z},$ and $W(R)((x))/N$ is complete with respect $v$. Hence $F(W(R))=W(R)((x))/N$ is maximally complete.