1. Do you imagine it as an 'L' shape like the following:

X

X

X X

2. Or do you think of it in a diagonal form like this:

X

X

   X

Just curious.  When looking at the board I always visualise the Knights movements like in the second case.

i imagine the possibilities of the next move.  so its like the knight is jumping straight to the square and doesnt have to abide by rank or file movement.

  

 Neither, I see the knight's move possibilities as a circle or starburst.

Like beyondwithin--jumping straight to the target.  The board's light and dark squares help a lot--knight goes to a different color each move.  Players are more surprised by the knight on its second move (forking possibilities, e.g.), so it's always necessary to check where it's going especially in the middle game.

Check out the famous Knight Tour Problem (with animation) http://thechessworld.com/knightour.html

 I think it will help you to visualise the knigt's moves. 

Usually in disgust, after the fact, when the D@#$ horse has run off with the Queen again.

I can see the eight initial squares - my mind no longer has to travel the 2 and 1 patterns.  And, thanks to Silman, I now look for paths for my knights to achieve good squares, and this affects my decisions in my development, which makes me make stronger move choices, sometimes being more subtle and patient (thanks Silman).  Since I look for paths to achieve good knight places, I eliminate more paths from my thoughts, unless I see that I can create a good path - then I look into it.  Silman's book How to Reassess Your chess is a wonderful book for helping you understand knights.  I have always liked knights better than bishops (although I have been working on improving my bishop play) - so I simply try to make good posts for my knights.

for futher knight exercises check out my blog 'that elusive knight'

http://blog.chess.com/PerfectGent/that-elusive-knight

 

I used to see them as L movements, but I agree with some of the other posts, I now more or less just know where I cna go without really thinking.