Hey! We use FORGET in Present indefinite sentences. For example someone has a problem of forgetting things, in this case we could say: He forget things too often. In your case #1, one just get out of his/her house; in this case we use 3rd form of verb (forgotten) so the sentence will be: She has forgotten to bring her purse. (completed action) case #2 is also similar, you have forgotten something and it doesn't matter that you are talking about just now or sometime in the past, because you got it that you have forgotten something (right now). We use FORGOT in Past indefinite tense. For example you were in Australia 2 years before and now you are talking to someone about that trip, in this case you can say: I forgot to brought my notebook when I was in Australia. I hope that this will help you.

Forget is used in present tense, and forgot is for the past. I'll rewrite your example sentences. Hypothetical situations: 1. You have just left your house, but have forgotten to bring your purse. 2. Arriving at the workplace, you notice you have forgotten to bring important documents. You do need present perfect here, as you are describing your present situation based on past events.