I can't speak for AA but the FAA does allow this (obviously if other companies have them) BUT the baby has to be removed in turbulence.

To be honest, I'm not sure if that would be even that useful. There is hardly any floor space, even at the bulkheads. Getting her down and out of that thing would be a bit awkward to say the least. You would have to pick her up every time it gets bumpy.

Are you flying alone?

A better option would be to bring an infant car seat and try to get it on the plane. Ask at check-in how full the flight is and if not, if you can have an empty seat placed next to you. This is really, the only safe way to fly with an infant.

Also, it would be much better to have a good baby carrier, whether you get the seat or not. Don't get an awkward "front pack" like a Bjorn. They're difficult to sit down with and the baby is held in a bad position for both of your backs. Get something like a wrap or sling. These can also be used as blankets, baby changers and breastfeeding covers while you're traveling. They do have a learning curve but especially for such a long flight, it would be well worth it.

You can also use carriers like a Beco or Ergo but for newborns, a sling is so easy to put on and take off. Make sure your baby is NOT attached to you for take-off and landing. It's dangerous and this is also a FAA reg. You can keep the baby in the carrier, like a blanket or whatever. Just detach it from you.

During the flight, having your baby comfortably attached to you will make your flight much, much easier, especially if you're alone. I've done 11 1/2 hour flights on my own with an infant and two other children and it was so great to just be able to get up, move around, etc. and not worry about her being left alone or having to hold her in my arms. My hands were free. I almost forgot she was there. It's possible to eat but be careful with hot food (I simply waited until it cooled down a bit).

Pick a portion of the flight when the F/A's aren't too busy and they'll be happy to hold your baby for you while you use the restroom.

At this tender age, they don't do much and quite frankly, are very easy to travel with, so there's your good news. They just want to be with their Mommies so putting them in a comfortable position while you watch the movie and having an option where you can get up and move around is really your best bet, IF you don't get an empty seat (good luck with that!!)