I'm reading Washington's book on cyclotomic fields, and he mentions that it is sometimes convenient to embed $\mathbb{C}_p$ into $\mathbb C$ and vice versa. In my mind, $\mathbb C_p$ and $\mathbb C$ seem like fundamentally different objects, so I'm wondering what such embeddings would look like, or how to think about this. In particular, when speaking of things like the Teichmuller character $\omega: (\mathbb Z/p\mathbb Z)^\times\rightarrow \mathbb C_p^\times$, which maps $a\in(\mathbb Z/p\mathbb Z)^\times$ to the unique root of $X^{p-1}-1$ in $\mathbb Z_p$ reducing to $a$ mod $p$, how can we consider $\omega$ as a complex character? Washington says that the Teichmuller character "may be regarded as coming from a complex character if desired, but the choice is noncanonical and depends on an embedding of $\mathbb Q(\zeta_{p-1})$ into $\mathbb Q_p$". Is the idea that we'd define the (complex) Teichmuller character to just send $a$ to a complex root of $X^{p-1}-1$. (Which root? I'd guess fixing $\zeta=\zeta_{p-1}$ primitive then setting $a\mapsto \zeta^a$?). This would then lead us to define the $p$-adic character by embedding $\mathbb Q(\zeta_{p-1})$ into $\mathbb Q_p$... Is this the right idea?
EDIT: My definition of the complex Teichmuller character doesn't work, as $\omega (ab)\neq \omega(a)\omega(b)$. But, if we send a generator of $ (\mathbb Z/p\mathbb Z)^\times$ to $\zeta_{p-1}$ then this should work.
As you argue yourself at the very end: Let $K$ be any field whose characteristic does not divide $p-1$ and in which the polynomial $X^{p-1}-1$ splits completely (i.e. contains the full $p-1$-th roots of unity $\mu_{p-1}(K)$). Then there are $\phi(p-1)$ (totient "$\phi$") distinct group monomorphisms $(\Bbb Z/p\Bbb Z)^\times \hookrightarrow K^\times$, namely, for every choice of a primitive $\zeta_{p-1} \in \mu_{p-1}$, there is exactly one which maps $1$ to $\zeta_{p-1}$.
That is just a fancy restatement of the elementary algebra fact that under the given conditions, $\mu_{p-1}(K)$ is cyclic of order $\phi(p-1)$, and needs no axiom of choice to summon non-explicit isomorphisms $\Bbb C \simeq \Bbb C_p$ or non-explicit automorphisms of $\Bbb C$.
If one wants to do that for $K=\Bbb C$, of course one can evoke those things and call all these maps "Teichmüller maps", but since they are virtually indistinguishable, and the AOC-dependence of those constructions makes it impossible to get any extra information to distinguish them, i.e. to tell us which of them should correspond to the unique Teichmüller map $(\Bbb Z/p\Bbb Z)^\times \hookrightarrow \Bbb C_p^\times$, I fail to see the benefit of this over treating it with elementary field theory as above. And I think that is also Washington's point when he says that the choice is noncanonical.