A problem is (Promise)-BQP complete if:
- The problem is in Promise-BQP, and
- The problem is Promise-BQP hard.
In general the reduction in (2) often uses Feynman-Kitaev clocks and looks very similar to those in Feynman's original outline showing Hamiltonian simulation is Promise-BQP hard, as discussed by @DaftWullie here.
For example, one fun Promise-BQP problem of Janzing and Wocjan involves estimating a number of $m$-length classical discrete-time random walks along an implicitly defined large graph that start and end at the same home position. But, the proof of the Promise-BQP Completeness in their paper is fundamentally a quantum reduction and looks strikingly similar to the granddaddy problem of Hamiltonian simulation. That is, after showing that their problem is in BQP through careful control of various errors in their algorithm, the authors go on to reduce Hamiltonian simulation to the problem of estimating the diagonal entries in a large matrix. Figure 1 in their paper used in the reduction, is certainly quantum.
Indeed I can imagine a hypothetical reduction from a problem $p$ to a problem $p_0$ that has a subroutine involving factoring large numbers a polynomial number of times. Certainly using factoring, which is not known to be classically polynomial but is in BQP, does not preclude the problem so reduced that just uses factoring as a subroutine from being Promise-BQP complete (although factoring itself is not likely to be BQP complete).