We know that completion of the field of rationals $\mathbb{Q}$ is the field $\mathbb{R}$ of reals with respect to usual metric.
Next, algebraic closure of $\mathbb{R}$ is the field $\mathbb{C}$ of complex numbers.
My question:
Does there exist an intermediate step of completion of the algebraic closure $\bar{\mathbb{R}}$ to get the field $\mathbb{C}$ ?
Because,
I understand the matter. But actually in p-adic field $\mathbb{Q}_p$ is the completion of the rational $\mathbb{Q}$ with respect to the p-adic absolute value and algebraic closure of $\mathbb{Q}_p$ is $\bar{\mathbb{Q}}_p$. Then completion of $\bar{$\mathbb{Q}}_p$ with respect to p-adic absolute value is the field $\Omega$ which is analogous to $\mathbb{C}$.
So can I say that trivially, algebraic closure of $\bar{\mathbb{R}}$ is $\mathbb{C}$ ?