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possible, but it sounds not very natural, difficult to think of the context.

я спрашиваю себя sounds like 'inner dialogue', dialogue with oneself, reflexive thinking.

спрашивать себя - ask myself smth.

I think this is the only meaning possible.

literally 'I wonder' means Мне интересно, but we dont' use such structures in Russian when asking stanadard questions like 'how are you, how u feel',

it's just better to use standard Russian...