The most common and most likely answer to this crossword clue is the 4 plus 8 letter word SELF PORTRAIT
I've seen this crossword clue in Financial Times
S
E
L
F
P
O
R
T
R
A
I
T
Buddy explains!
The straight part
We know that "profile possibly, egoistic one" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case profile possibly, egoistic one = self portrait.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does SELF PORTRAIT mean?
SELF PORTRAIT noun- The bust of an artist by the artist him/herself; a portrait of the painter who painted it.