Recently the separation between BQP and PH has been proven. Does this result tell us something about the relation between BQP-complete problems (e.g. approximation jones polynomial solved by AJL algorithm) and BQP\PH? In particular, is the latter included in the former?
Indeed I think it follows from (1) showing that evaluating the Jones polynomial is (Promise)BQP-complete, and (2) the existence of an oracle separation between BQP and the polynomial hierarchy PH, that (3) evaluating the Jones polynomial is not likely to be in PH. That is, evaluating the Jones polynomial is precisely in BQP\PH as you had proposed.
In detail, Raz and Tal had shown that the forrelation problem of deciding whether two Boolean functions $f$ and $g$ given by black-boxes are "forrelated" (e.g., whether the Fourier transform of $f$ is correlated with $g$) is, as an oracle problem, outside not just P, not just NP, not just AM, but even outside the entirety of the polynomial hierarchy PH. But, previously Aharanov, Jones, and Landau had provided an algorithm showing that evaluating the Jones polynomial is PromiseBQP-Complete.
Thus, because the Jones polynomial is BQP-Complete, there is a reduction from any problem in BQP (such as forrelation) to the evaluation of some Jones polynomial. Hence, we can relate forrelation to this evaluation, and use AJL's BQP algorithm to evaluate the Jones polynomial as a vehicle to evaluate the forrelation, but we can't evaluate the forrelation in PH because that was ruled out by Raz and Tal. Thus the particular Jones polynomial related to forrelation cannot be evaluated in PH, and therefore evaluating the Jones polynomial is in BQP but not in PH.
This is all rough and imprecise intuition about reductions relative to oracle separations, and also with respect to promises about the gap between the forrelations of $f$ and $g$ and the accuracy of the Jones polynomial evaluation. For me at least these require some effort to be particularly correct about, and I often get tripped up into the precise way to make the statements!