Suppose that $\{x_n\}$ is a Cauchy sequence under the ultrametric $|\cdot|$, and consider the sequence $\{|x_n|\}$ of real numbers.
If the sequence is eventually constant, then there is nothing to do: the absolute value of the limit is equal to the absolute value of some element of $K$.
If the sequence is not eventually constant, then we can find a subsequence in which $|x_{n_k}|\neq|x_{n_{k+1}}|$ for all $k$. Since a sequence is Cauchy in an ultrametric if and only if the sequence of consecutive differences $|a_i-a_{i+1}|$ goes to $0$, and we have $|x_{n_k} - x_{n_{k+1}}| = \max\{|x_{n_k}|,|x_{n_{k+1}}|\}$ (since the norms of $x_{n_k}$ and of $x_{n_{k+1}}$ are different), then we must have $\lim_{k\to\infty}|x_{n_k}| = 0,$ and hence, since the original sequence is Cauchy, $\lim_{n\to\infty}|x_n| = 0.$ Therefore, the absolute value of the limit is $0$, which is the absolute value of some element of $K$ as well.
In either case, the absolute value of the limit of a Cauchy sequence of elements of $K$ is always equal to the absolute value of an element of $K$, so the set of absolute values of elements of the completion is equal to the set of absolute values of elements of $K$.