The $p$-completed algebraic $K$-theory of the algebraic closure of $ Q_p$, i.e., $K(\bar Q_p; Z_p)$, is equivalent to its second loop space, up to an issue about path components. This is due to Suslin. The descent to $Q_p$ is more subtle than the descent from $C$ to $R$, because the absolute Galois group of $Q_p$ is much more complicated than that of $R$. Still, if you reduce to homotopy with $Z/p$ coefficients, $K(Q_p; Z/p)$ is equivalent to its $(2p-2)$-fold loop space, up to the same issue as before. B{"o}kstedt and Madsen proved this using topological cyclic homology. I did the case $p=2$. Later it followed from the proof of the Lichtenbaum--Quillen conjectures by Voevodsky and others.

The $p$-completed algebraic $K$-theory of the algebraic closure of $\mathbb{Q}_p$, i.e., $K(\bar{\mathbb{Q}}_p; \mathbb{Z}_p)$, is equivalent to its second loop space, up to an issue about path components. This is due to Suslin. The descent to $\mathbb{Q}_p$ is more subtle than the descent from $\mathbb{C}$ to $\mathbb{R}$, because the absolute Galois group of $\mathbb{Q}_p$ is much more complicated than that of $\mathbb{R}$. Still, if you reduce to homotopy with $\mathbb{Z}/p$ coefficients, $K(\mathbb{Q}_p; \mathbb{Z}/p)$ is equivalent to its $(2p-2)$-fold loop space, up to the same issue as before. Boekstedt and Madsen proved this using topological cyclic homology. I did the case $p=2$. Later it followed from the proof of the Lichtenbaum-Quillen conjectures by Voevodsky and others.