I wonder the existence of non-isomorphic models of $\text{Hen}_{0,0}$ with isomorphic RV sorts.
To clarify the notation, $\text{Hen}_{0,0}$ means Henselian valued field with equicharacteristic $(0,0)$, which means that both the valued field and its residue field have characteristic $0$.
The question comes from AKE principle. As a famous example, $\mathbb{Q}_p$ and $F_p((t))$ have the same valued field and residue field. Also, in these two fields, the angular component map (for short, ac-map) is built-in by taking the coefficient of the "leading term" in the power series representations (viewed as elements in local fields!). Now, $\mathbb{Q}_p$ and $F_p((t))$ have the same RV-sort, because the existence of ac-map witnesses that the exact sequence $1\to k^\times\to RV^\times\to \Gamma\to 0$ splits. However, those two fields are not isomorphic.
It was attractive to think that Henselian fields with equicharacteristic $(0,0)$ are isomorphic, whenever they have the same $k$ and $\Gamma$. This is actually wrong. Take a model of Hen$_{0,0}$ that is not maximal (a concrete example?) and look at its maximal extension, which exists by Krull's theorem. Then they have the same $k$, $\Gamma$, but are not isomorphic, because otherwise, say, the dimension of $L/K$ should be the same as that of $L/\sigma(K)=0$, contradiction. However, we don't really have ac-map now, so they may not have the same RV sort, even if the $k$, $\Gamma$ are the same. It also doesn't work by looking at $\aleph_1$-saturated model, because we cannot control existence of ac-map and being an immediate extension.
To sum up, here are two of my questions:
Is there a model of Hen$_{0,0}$ that is not maximal, i.e. a valued field of equicharacteristic $(0,0)$ that is algebraic maximal but not maximal?
Are there non-isomorphic models of Hen$_{0,0}$ with isomorphic RV sorts?