Botkin's diagnosis of Alexandra being 'not normal' was the sort of thing men were always diagnosing about women in the 19th century. The other diagnosis was that of hysteria, which Botkin also attributed to Alexandra. It was a form of attempted control, particularly for women in positions either of power or perceived power. Alexandra, as the wife of the Russian autocrat, self-secluded from any other close contacts except her immediate family, was particularly subjected to this sort of judgement, not just by Botkin. However, a very similar diagnosis was made of her grandmother, Queen Victoria, by her husband Prince Albert, who was frightened of her emotionalism and wanted more political control. In both cases these sorts of judgements were made by well-meaning men who nevertheless were not in the positions of control as they would have expected normally to be, and sought to quite literally rationalise the situation by such views. I think Alexandra was obstinate, tactless, politically naive, and prone to emotional judgements, but I haven't seen anything which convinces me she was actually insane. As for 'not normal', however many people may have been writing behind Alexandra's back that she was not normal, or hysterical, or just a nutcase, there wouldn't have been a single person who would have said it to her face, except in a roundabout, respectful sort of way which she could brush aside. And that, of course, just make her feel that what she did was right and she shouldn't bother with views other than her own, however stupid or actually insane she may have appeared to others.
