For any field of $p$-adic numbers $\mathbb{Q}_p$ there is a unique, up to an isomorphism, extension $\mathbb{C}_p$ that is both algebraically and metrically complete, the field of $p$-adic complex numbers. Further, there exists a unique spherically complete extension $\Omega_p\equiv \mathbb{C}_p^{sph}$ of $\mathbb{C}_p$, whose elements are easily characterized:
The elements are exactly those power series $\sum_{r \in \mathbb{Q}} a_r p^r$ with coefficients given by Teichmüller representatives of $\overline{\mathbb{F}_p}$, such that the set of exponents with nonzero coefficients forms a well-ordered subset of the rationals.
(see the post https://sbseminar.wordpress.com/2007/08/21/p-adic-fields-for-beginners/, and Poonen's article http://www-math.mit.edu/~poonen/papers/amsval.pdf)
The question:
Is there any alghorithm known that produces a power $p$ series, like above, for the roots of a given polynomial $f\in\mathbb{Q}_p[x]$ ?
The situation is more or less clear when equation $f(x)=0$ is solvable in radicals, e.g. $f(x)=x^2-3 x +1 \in\mathbb{Q}_5[x]$ has a root $$x_1=\frac{3+\sqrt{5}}{2}= 4 + 3\cdot 5^{1/2} + 2\cdot 5^1 + 2\cdot 5^{3/2} + 2\cdot 5^{2} + \ldots\; \in \mathbb{C}_5 .$$ But what about general case?
UPD. Elements $x\in\mathbb{C}_p$ inside $\mathbb{C}_p^{sph}$ can be quite easily characterized: for any rational $t\in\mathbb{Q}$ there are only finitely many $a_r\neq 0$ with $r<t$. To see this one should approximate $x$ by $y\in\mathbb{Q}_p^{alg}$ such that $|x-y|_p < p^{-t}$. Since $y$ belongs to some finite extension of $\mathbb{Q}_p$ with valuation group of the form $\{ p^ {-k/n} : k\in \mathbb{Z}\}\subset(0,+\infty)$, $n$ fixed, the expansion $y=\sum_{r\in\mathbb{Q}}{b_r p^r}$ can have nonzero $b_r$ only for $r=k/n$. Since $a_r=b_r$ for $r<t$, the statement follows.
UPD2. Numerically finding roots from coefficients is an ill-defined problem for polynomials over $\mathbb{C}$, as an example of Wilkinson's polynomial shows (see https://en.wikipedia.org/wiki/Wilkinson%27s_polynomial). I guess, the situation is as bad over $\mathbb{C}_p$. Or not?