Theorem 1.2 of the paper "The motivic Hopf map solves the homotopy limit problem for K-theory" (see <https://elibm.org/article/10011880>) says that (under certain assumptions on the base field) the homotopy fixed points of the action of $C\_2$ on the motivic algebraic $K$-theory spectrum $KGL$ give the $\eta$-completion of the Hermitian $K$-theory spectrum $KQ$, where $\eta$ is the (Morel's) algebraic version of the Hopf map. My question is: is it known whether $KQ$ is $\eta$-complete in this context, i.e., whether the $\eta$-completion operation does not really change anything? 



Is the motivic homotopy spectrum of Hermitian K-theory $\eta$-complete?
If I understand the question correctly, you're asking if there are base fields $k$ of characteristic not $2$ and such that $k[\sqrt{-1}]$ has a finite $\mathbb{Z}/2$-cohomological dimension (the assumptions used in the above mentioned paper) for which the map $KQ \to KGL^{hC\_2}$ is not an equivalence. Note that by the Tate fiber square (mentioned also in that paper), this is the same as asking whether $KW \to KGL^{tC\_2}$ is an equivalence, a necessary condition for which is that the map $W(k) \to \pi\_0(K(k)^{tC\_2})$ is an isomorphism of abelian groups, where $W(k)$ is the Witt group of $k$ and $K(k)$ is its $K$-theory spectrum. As shown in Theorem 16 of [this paper](https://www.sciencedirect.com/science/article/pii/S0001870811001691), the last map coincides in this context with the completion map 
$$ (\*) \quad W(k) \to \lim\_n W(k)/I^n $$ 
with respect to the augmentation ideal $I \subseteq W(k)$. An example of a field satisfying the above conditions and for which this completion map is *not* an isomorphism is the field $\mathbb{R}$ of real numbers, where $(\*)$ is the map $\mathbb{Z} \to \lim\_n\mathbb{Z}/2^n$.


