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?

Algebraic $p$-adic expansions, J. Number Theory 23 (1986), no. 3, 279–284 (MR). $\endgroup$ – LSpice Nov 17 '16 at 17:14