Of Monsters and Men are best known for a lighter folk-pop sound but "Alligator" finds the Icelandic band utilizing a heavier kick than usual.

The driving, percussion-heavy track was penned by co-vocalist Nanna Bryndis Hilmarsdóttir. She recalled to NME that it came very deep into the writing process for the Fever Dream album. Usually she'd write on an acoustic guitar and guitarist Raggi Þórhallsson would "flesh it out in a very singer-songwriter kind of way." However she'd become tired of this method of writing and was looking for ways to challenge herself.For this song, she did a lot more writing on the computer and explored sounds there.

Hilmarsdóttir found herself writing more on her own for Fever Dream compared with previous Of Monsters and Men releases, which gave her lyrics more of a female perspective. This song is a case in point:

Shake it out
It's just what I'm feeling
And now I take control
I'm fever dreaming

Hilmarsdóttir explained she "wanted to feel strong. It was about finding that space of feeling empowered. You can hear it in the music - it's taking space."

The "Alligator" title of the song does not appear in the lyrics. It's likely based on certain lines such as the singer "laying in a stream."

The song was produced by producer Rich Costey (Muse, Rage Against The Machine, Death Cab For Cutie). According to Hilmarsdóttir, Costey pushed the band towards the more "bombastic" and "out-there" edges of their sound for Fever Dream.

The song was performed on TV for the first time during Of Monsters and Men's appearance on the May 15, 2019 episode of The Tonight Show Starring Jimmy Fallon.