Minimum number of clues to fully specify any sudoku?
Since permuting two rows within a single block of a valid, completed sudoku produces another valid, completed sudoku, you can take any completed board (81 clues) and remove the first two rows (81-18=63 clues), which will give you an incomplete sudoku with two solutions. Note that even if you remove all but one of the 18 numbers there, the solution is immediately uniquely determined (since there can be no number repeated in the same column).

Another operation that produces another completed sudoku is applying a permutation of $\{1,\ldots,9\}$. If you take a permutation that is a transposition (permutes two elements, keeping the other fixed), again, as before, you can remove all appearances of those two elements and you have an incomplete sudoku with two possible solutions and 63 clues. Again, if you don't remove all 18 numbers, the solution will be unique.

Of the six elemental operations that produce a completed sudoku (see here), these two are the ones that can involve the least number of elements, so I'd say $C=63$ is an upper bound for what you're looking for. I know this does not exactly answers your question, but the general idea of removing sets of positions which produce two different solutions might be a good starting point.