Classical complexity theory makes much of the study of so-called intermediate problems - that is, problems that are in $\mathsf{NP}$ but are nonetheless not known to be in $\mathsf{P}$ and further not expected to be $\mathsf{NP}$-complete.
Commonly discussed examples of likely intermediate problems include, e.g., GRAPH-ISOMORPHISM and FACTORING. One reason why GRAPH-ISOMORPHISM and FACTORING are both thought to be of intermediate complexity is that neither are known to have an algorithm in $\mathsf{BPP}$, while both are known to be in $\mathsf{coNP}$ (or rather, $\mathsf{coAM}$ in the case of GRAPH-ISOMPORPHISM) - thus if, for example, FACTORING/GRAPH-ISOMPORPHISM were $\mathsf{NP}$-complete, then one would have the conclusion that $\mathsf{NP=coNP}$/the polynomial-time hierarchy $\mathsf{PH}$ collapse, both of which are believed to be unlikely.
Turning now to the $\mathsf{BQP}$ complexity class, noting the fact that FACTORING is both in $\mathsf{NP}$ with witnesses being the factors, and in $\mathsf{BQP}$ by Shor's algorithm, a conclusion is that FACTORING is not likely to be (promise) $\mathsf{BQP}$-complete. If FACTORING were complete for $\mathsf{BQP}$ then, for example, $\mathsf{BQP}\subseteq\mathsf{NP}$, which may be ruled out by the recent breakthrough of Raz and Tal.
Thus it may be natural to ask:
What is stopping FACTORING from being complete for $\mathsf{BQP}$?
GRAPH-ISOMORPHISM can be generalized/altered to SUBGRAPH-ISOMORPHISM, which is $\mathsf{NP}$-complete. Can FACTORING be generalized or altered in any way such that it is complete for $\mathsf{BQP}$, in much the same way that SUBGRAPH-ISOMORPHISM generalizes GRAPH-ISOMORPHISM to be $\mathsf{NP}$-complete?
