Great background, tack sharp, great exposure, good head angle - what more could you want? I don't mind seeing only the left eye, especially with the profile of the beak clear, not overlapping other structures. Was there any more room at the bottom?
Being a background person, this one passes in a big way! As far as the re-cropping thoughts/comments... just think eye placement and rule of thirds right? So maybe a small amount off the left side would be good, but thats all personal preference stuff.