Alex says Yuki-Onna said she'll kill him if he ever breaks his vow not to tell anyone about their meeting. Actually, she never says she'd kill him, just that she'd know if he broke his vow.
Though Alex is living out a myth (i.e., not real life or history), it nonetheless seems suspicious that he's one of two men who go off into the woods together where one dies and the other lives. He never returns to the village nor has to explain the circumstances of his mentor's death, yet is never suspected of foul play.
Alex is surprisingly and uncharacteristically unsympathetic and dismissive to Mosaku's widow, the loving wife of a caring man who was Minokichi's/Alex's mentor.