The quest must be active (unsolved) by the person landing the killing blow.
Not true. In making "sirs" to enable hard core, the phoney "sir-to-be" will get the credit even if the shepherd has already completed the quest and does the actual killing. This we've done a few times.
It doesn't always work, and I haven't tried it often enough to completely pin it down, but it seems that, if the "sir-to-be" is on the level where the boss dies and does some damage to the boss, then he'll get the credit no matter who strikes the final blow. I do know that not one of those "sir-to-be" struck the killing blow (in one case, he was dead when that happened). I don't know why it failed in the cases it did, but that was only with Diablo (not the other act end bosses), and I suspect "sir-to-be" did not do any damage.
