Early in the novel, as we're getting to know Baba, Amir relates one of the legends about his father. Apparently Baba wrestled a black bear in Baluchistan and has the scars to prove it. Now, we know what you're thinking: no one could wrestle a bear and live to tell the tale. But Amir reassures us this story isn't typical Afghan laaf (exaggeration). The story has obviously affected Amir because he imagines it "countless times" and even dreams about it (3.1). And here's the interesting part. In his dreams, Amir can't tell Baba apart from the bear.
On one level, you can interpret the bear story fairly simply: it tells us just how towering of a figure Baba is to Amir. This guy is no joke – he wrestles bears. The fact that Amir believes the story, too, tells us a little about their relationship. It's one of distant awe. But there's also the oddity of the bear and Baba morphing into each other. Perhaps Baba becomes a fearful beast to Amir? Or perhaps Baba, in wrestling with his sins, merges with them? We're not totally sure. But Baba and Amir both wrestle with major betrayals in the book. At one point Amir explicitly compares the troubles and hardships of Baba's life to a bear Baba couldn't beat (see 13.51).
Later in the novel, when he's in the hospital in Peshawar, Amir has a hallucination. In the hallucination, Baba is in Baluchistan fighting the bear. It's a rip-roaring fight. Fur flying and all that. When the dust clears, Amir gets a good look at the person wrestling the bear. It's not Baba – it's Amir. Now Amir has taken on the fight with the bear. Does this mean Amir achieves some sort of manhood – or only that he's taken on his father's sins? (They could be the same thing.) If we keep in mind the earlier dream in which Amir can't tell his father apart from the bear, an interesting interpretation pops up. Perhaps when Amir wrestles with the bear he is really wrestling with his father. We'll say this for Amir and Baba – Hosseini has them wrestling with one of the most dangerous animals on the planet. We would choose a cuddly kitten or maybe a lame gerbil.