Four examples I could think of, off the top of my head:
1. As noted in the comments, DC has a character named the <PERSON>. Her alter-ego is <PERSON>.
The <PERSON> is portrayed as either a full-fledged case of dissociative identity disorder, or something akin to that. Originally, when <PERSON> went to sleep, the <PERSON> identity would take control of her and hunt for criminals. I don't think multiple personality disorder (as it was known at the time) was explicitly referenced, but it was certainly implied.
2. There have been times when Marvel's Sentry, in his <PERSON> identity, has been unaware of his super-hero identity. However, in this case, he was not acting as the Sentry during this time (as the Sentry was too dangerous to exist, but none of those responsible were willing (or able?) to kill him). The processing of making him (and everyone else on Earth) forget his being the Sentry appears to have involved both magic and science (Dr. <PERSON> and <PERSON> of the Fantastic Four were definitely involved).
3. <PERSON>, the Captain America of 2099 shown in the 2015-2017 Spider-Man 2099 run, has no idea she's <PERSON>. 837
It appears that she underwent some sort of mental conditioning (done by <PERSON>, if done) that allows her persona to switch between <PERSON> and <PERSON>. I believe <PERSON> has at least some knowledge of her <PERSON> identity, but I'm not certain. For that matter, as there may be some degree of physical transformation as well (change in height or musculature).
4. DC's Firestorm had another take on this. Firestorm was formed by the fusion of two characters - <PERSON> and <PERSON>. When the nuclear accident that transformed them into Firestorm happened, <PERSON> was unconscious. Thus, when they formed Firestorm, <PERSON> basically was Firestorm's personality and basic knowledge base - but, <PERSON> was present in <PERSON>'s mind, and could communicate with him and advise him (useful - a high school student with average grades wouldn't always make the best choices when transforming matter at an atomic level).
However, when the two separated, <PERSON> originally had no memory or knowledge of what had transpired; as far as he could tell, he was having blackouts. After this issue came close to destroying his life, <PERSON> was able to convince him of what was going on, and he eventually would retain at least some memory of what they did as Firestorm. So, this was a temporary situation - but one that lasted well over a decade, in terms of the character's published adventures.
Interestingly, each of the above examples has a somewhat unique rationale for the separation of identities. I'm sure there are other examples. Some may echo one or more of the explanations above; some may introduce yet other methods to explain the situation. 663