Bruce Wayne, or Batman, lives in the face of danger. Fighting criminals in the shadows and facing super villains such as the Joker, Two Face or even Mr. Freeze. What did he fear? Did he get over his fear of bats?
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In Knightfall, Batman is dosed by the Scarecrow's fear gas in (Batman #496 KnightFall: Part 9 - "Die Laughing") in front of the Joker. It has the effect of making him hallucinate Jason Todd (Robin II) which drives him into a psychotic attack on the Joker, suggesting that Batman's fear is "loss", i.e. losing people like he lost his parents and how (at that time) he had lost Jason.
When he is dosed with the gas:
ROBIN! He's killing robin! No -- not real -- Jason's already dead -- just visions.. from the gas..
Somewhat spoiler, please be careful when you are reading this if you have not seen the movie...
Something that was mentioned more than one time was the fear of death. Bruce Wayne or Batman say that they (he) does fear death.
Bruce Wayne: Why didn't you just... kill me?
Bane: You don't fear death... You welcome it. Your punishment must be more severe.
Blind Prisoner: You do not fear death. You think this makes you strong. It makes you weak.
Bruce Wayne: Why?
Blind Prisoner: How can you move faster than possible, fight longer than possible without the most powerful impulse of the spirit: the fear of death.
Bruce Wayne: I do fear death. I fear dying in here, while my city burns, and there's no one there to save it.
Although that does not exactly answer your question, it is an important theme of the movie.
Bruce wayne also fears of himself. i.e., if ever he becomes the antagonist and harms the people. His character had spoken this dialogue while having conversation with superman, in justice league.