There are two "half past twelve's" every 24 hours, so yes, you must make it clear if the time is referring to the morning or afternoon. It may already be clear that it is AM or PM based on what has been written or said earlier (based on the context). If you are referring to the early morning, i.e. AM, then the sentence could be written as: - half past midnight, - half past twelve in the morning, - twelve thirty in the morning, or - twelve-thirty AM. If you are referring to the early afternoon, i.e. PM, then the sentence could be written as: - half past noon - half past twelve in the afternoon, - twelve thirty in the afternoon, or - twelve thirty PM. We never say "half past twelve at midnight".

You say: Half past twelve in the morning.