Following this question:

Given a valued field $K$, denote with $\bar{K}$ its algebraic closure and with $\hat{K}$ the completion. Then both $\hat{\bar{K}}$ and $\hat{\bar{\hat{K}}}$ are complete and algebraically closed. If $K=\mathbb{Q}$, these constructions always give $\mathbb{C}$ or $\mathbb{C}_p$ for a prime $p$, i.e. both constructions give the same fields.

Is this true for any field $K$? Or is there a field $K$ with a valuation $v$ such that for no extensions of $v$ to $\bar{K}$ we have $\hat{\bar{K}}=\hat{\bar{\hat{K}}}$?