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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.