For a start, the reasoning is incorrect:
Strictly homoeopathy means "treating like with like". That is to say, you treat an ailment with a medicine that would in other circumstances cause the same symptoms.
Generally though, by homoeopathy we mean something narrower still, which is the use of extreme dilutions.
But if we take the word at its root, we could argue that conventional vaccines are "homoeopathic" in this non-specific sense.
Allopathy is the exact opposite - treating something with something that causes the opposite effect. You could reasonably apply the term to some conventional approaches (running your finger under a cold tap if you've burnt it is an allopathic remedy!), but not to all of them.
If we take this broad approach to these terms, we can find ourselves applying either depending on how precisely we look. We could consider digitalis. Overdosing on digitalis can cause fatal heart disturbances. Do we consider its use to treat heart conditions homoeopathic (treating heart conditions with something that causes heart conditions) or do we note its antiarrhythmic effects are what cause such fatal heart disturbances and so use it to treat people with heart conditions that would benefit from those effects - essentially an allopathic use.
(Amusingly, digitalis is used by both conventional and homoeopathic practitioners to treat some heart conditions).
Most importantly though, not only are not all conventional treatments "allopathic" in this broad sense, but that is not how doctors and medical researchers consider their field. They do not look at a symptom and then immediately concentrate upon looking for various things in the word that cause the opposite to occur.
As such, allopathic is inappropriate to use to describe conventional medicine not just on conventional medicine's terms, but on Hahnemann's too.
I think it could be described as one of the following:
- display picture
- profile pic
- display image
- your pic
- personal photo
- profile photo
I would not use avatar as a synonym as that would reference what is usually not a picture of yourself, but an icon of some sorts. My avatar is a picture of rye bread, the stack exchange sites have a random icon generator for avatars. Some sites do a collection of robots or weird puppets. Again my opinion is that avatar usage is different from profile picture.
Best Answer
Perhaps genie is in keeping with your theme but different enough from wizard.
It has been used before in a similar sense and can perhaps be said to be an abbreviation e.g.
Otherwise guide and all its synonyms....