convergent series representation for p-adic complex numbers The field $\mathbb{C}_p$ of $p$-adic complex numbers is the completion of the algebraic closure of $\mathbb{Q}_p$ with the corresponding extension of the usual non-Archimedean valuation $|\;\;|_p$.
Every $x\in\mathbb{Q}_p$ has a unique representation of the form $\sum_{i=m}^\infty a_ip^i$, where $m\in\mathbb{Z}$ and the $a_i$'s are representatives of the classes in $\mathbb{Z}/p\mathbb{Z}$.
Does $y\in\mathbb{C}_p$ have a similar representation as a generalized power series?   Any reference? 
 A: The blog post http://sbseminar.wordpress.com/2007/08/21/p-adic-fields-for-beginners gives a good overview of the situation.
To briefly summarize (extracted from the above post — any errors are probably mine):


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*The elements of $\mathbb{Q}_p$ are exactly those represented by series of the form $\sum_{r \in S} a_r p^r$, where $S$ is a bounded-below subset of the integers and each $a_r$ is a multiplicative lift (i.e., a "Teichmüller lift") of an element of $\mathbb{F}_p$.

*The elements of $\mathbb{Q}_p^{nr}$ (the maximal unramified extension of $\mathbb{Q}_p$) are exactly represented by series of the form $\sum_{r \in S} a_r p^r$, where $S$ is again a bounded-below subset of the integers, but the $a_r$ can now be lifts of any element of $\mathbb{F}_q$, where $q$ is a fixed power of $p$. (As explained in the comments below, the $a_r$ cannot be lifts of any elements of $\bar{\mathbb{F}}_p$; they must all be contained in a common finite extension of $\mathbb{F}_p$. This constraint is removed in the completion of $\mathbb{Q}_p^{nr}$.)

*The elements of $\bar{\mathbb{Q}}_p$ and its completion $\mathbb{C}_p$ are represented by certain series of the form $\sum_{r \in S} a_r p^r$, where the $a_r$ are again lifts of elements of $\bar{\mathbb{F}}_p$, but now $S$ can be a more general well-ordered subset of $\mathbb{Q}$. Which such series give rise to elements of $\mathbb{C}_p$ is discussed in Kedlaya, Power series and $p$-adic algebraic closures (MR).

*The elements of $\Omega_p$, the spherical completion of $\mathbb{C}_p$, are represented exactly by series of the form $\sum_{r \in S} a_r p^r$, where the $a_r$ are as before, but now $S$ can be any well-ordered subset of $\mathbb{Q}$ (with no other restrictions). This is attributed to the undergraduate thesis of Bjorn Poonen, written up in Poonen, Maximally complete fields (MR).


And this is in some sense as far as we can go, since $\Omega_p$ is "maximally complete": we can't extend it any further without "adding geometry", in the sense that it's the unique largest field of characteristic zero with value group $\mathbb{Q}$ and residue field the algebraic closure of $\mathbb{F}_p$.
