The most common and most likely answer to this crossword clue is the 2 plus 2 plus 2 plus 4 letter word UP TO MY NECK
I've seen this crossword clue in The Guardian
U
P
T
O
M
Y
N
E
C
K
Buddy explains!
The straight part
We know that "Deeply involved/with head still above water" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case deeply involved/with head still above water = up to my neck.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.