Choosing an isomorphism $\tau:\overline{\mathbf{Q}}_p\simeq\mathbf{C}$; how do things depend on choice of $\tau$? I sometimes see arguments that begin by choosing an isomorphism of fields $\tau:\overline{\mathbf{Q}}_p\simeq\mathbf{C}$, and then defining some property in terms of this isomorphism. I'm not so familiar with the technical properties; must there exist continuous isomorphisms or isometries? Where can I read up on the basic properties of such isomorphisms or how this technique is deployed?  
Sample question: suppose I say that $P\in\mathbf{Q}_p[X]$ is 'pure of weight $i\in\mathbf{Z}$' if every root $\lambda\in\overline{\mathbf{Q}}_p$ of $P$ satisfies $|\tau(\lambda)|_{\mathbf{C}}=i$. Does this notion depend on $\tau$?
 A: Your notion of purity does not depend on $\tau$ for polynomials with $\mathbb{Q}$ coefficients but does depend on $\tau$ for arbitrary polynomials with $\mathbb{Q}_p$ coefficients. Any two isomorphisms from $\overline{\mathbb{Q}_p}$ to $\mathbb{C}$ will differ by an automorphism of $\mathbb{C}$. An automorphism of $\mathbb{C}$ must interchange roots of any polynomial with $\mathbb{Q}$ coefficients, so if $P$ is pure with respect to one choice of $\tau$ it is pure with respect to an arbitrary $\tau$.
To see that it doesn't work in general, let's work over $\mathbb{Q}_7$. This field has two square roots of 2(Hensel's lemma); we call them $\pm \alpha$. We'll reserve the symbol $\sqrt{2}$ to mean the usual positive element of $\mathbb{R}$. There's a $\tau$ that takes $j$ to $\sqrt{2}$ and there's a $\tau$ that takes $j$ to $-\sqrt{2}$ (simply compose $\tau$ with any extension of the unique automorphism of $\mathbb{Q}(\sqrt{2})$ to $\mathbb{C}$. Now consider the polynomial 
$$
(x-1)^2 - \alpha.
$$
If $\tau(\alpha) = \sqrt{2}$ then the roots of this polynomial are $1 + 2^{1/4}$ and $1 - 2^{1/4}$, which have different complex absolute values. If $\tau(\alpha) = -\sqrt{2}$ the roots are $1 + 2^{1/4}i$ and $1 - 2^{1/4}i$, which have the same complex absolute value.
