Should I Purify Shadow Ho-Oh?

If you're just in it for the battling, don't Purify. If you want something unique that looks stunning in your storage and in AR photos while still being a powerhouse, Purify. Personally, I already have a standard Shadow Ho-Oh, so it's no question.

Is it worth it to purify apex shadow Ho-Oh?

If you have any interest in battling and core Pokemon Go gameplay, however, there is no reason to purify your Apex Ho-oh or Lugia. The 'boosted' attacks are a myth, and you'll kick yourself for enacting an irreversible and ill-advised change to these one-off Legendaries.

Should I purify my shadow legendary?

Should you purify Shadow Legendaries? Due to the limited amount of opportunities you have to encounter Giovanni and catch his shadow legendary Pokémon, almost any shadow legendary you capture should never be purified. It's better to keep them in their shadow versions, and the same goes for shadow Ho-Oh.