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After listening to Liam Neelson's voice for ten years, Anakin wouldn't be the brat we see in AOTC.

Seriously though, since Qui-Gon had a romance with Tahl that pushed him to the brink of the Dark Side, he would know what uncontrolled passion could do to a Jedi.

Logically, he would notice what was happening with Anakin and prevent him from developing these thoughts. He would share his experiences with Anakin and encourage him to become less selfish.

But of course, in the PT, events just happen for the sake of the plot getting somewhere, as opposed for a reason. Therefore, even if Qui-Gon became Anakin's master, it probably wouldn't have mattered. Obi-Wan did fine- Anakin, a 20 year old Jedi, acted as mature as a 12 year old in AOTC.