Do such field automorphism of $\mathbb{C}_p$ exists?

Question

Let $\mathbb{C}_p$ be the p-adic complex field and $ \{a_i\}\subseteq \mathbb{C}_p$ be a finite set, then do there exist a field automorphism $\phi:\mathbb{C}_p\to\mathbb{C}_p$ such that for all $i$ we have $\phi(a_i)\in \mathbb{\bar{Q}_p}$?

I guess Zorn's lemma might be helpful.