As a former pilot...buy a seat. Is the risk of something happening (even minor like turbulence) worth it? I did it once when my DD was 9 months and bought a seat for my DD 1 year. I held her so tight the 3 hours, that I was in pain when we landed. There was a seat available on the way home so we used that with her carrier.
The seat does have to be by a window so that it doesn't block anyone and it has to be FAA approved. My husband and I have to sit in 2 different aisles because we have 2 car seats. Plus, they both fall asleep upon take off for the entire flight.