When Aang accidentally creates blue flames he chalks it up to being the Avatar, master of all four elements. Then Katara brings up double bending.

Double benders, as they're called, are soulmates who share bending abilities. They are so rare that they were considered a myth 100 years ago, and are largely still considered to be one by many today.

Katara doesn't think it's a myth. Sokka points out that Azula could have potentially escaped with airbending.

Potentially.

Then Zuko asks if double bending would mean Azula shares airbending with Aang or if she shares all of the elements he can bend, and Aang doesn't have an answer. Azula's escape has gone from unfortunate to potentially world changing.

Potentially.

Aang decides to contact his past lives via Avatar state to double check that there isn't some spiritual double bending loophole that would allow Azula to bend all four elements.

Aang learns he is the first and only double bending Avatar.

Aang can now both redirect and shoot lightning, which is pretty cool, but he has noticed that he can redirect lightning better than Zuko; and he wonders whether that's because he's the Avatar or because somewhere, somehow, Azula has learned to redirect lightning.

Aang can also make his fire blue. He demonstrates this in a fireworks show to his friends. They applaud at all the right times, but Aang notices the fear in their eyes so he sticks to regular fire. Of course, they notice he noticed, so his friends gather one day to tell him that it's okay if he uses the blue flames and that if Azula can benefit from airbending then he can benefit from having a flame that burns hotter. Aang thanks his friends, but he still avoids the blue.

It's not because he's scared of her (Aang will freely admit he's scared of Azula. Even her father didn't make his heart race as much) but because the blue flames remind him of the soulmark, scar, mark on his back.

He doesn't know what he'll say or what he'll do when they find her. No one knows if Azula has gained the powers of an Avatar, but Aang is the only one who questions whether she would use them.