Is Anapanasati a complete path? Does it, by itself, lead to nirvana?
Initially the Bodhisattva was practicing a form of Pranayama in the form of holding or controlling one's breath. [Deva Sangarava Sutta, Maha Saccaka Sutta]

The Breath meditation practiced later was was observation of phenomena. Each 4 Sathipattana is covered in sets of 4 Tetrad with a total of 16 steps. Also it is the complete path. The 16 steps can be found in Anapanasati Sutta and Anapanasati