The answer to What are quaternions and how are they used to represent spacecraft dynamics? gives us a thorough overview of the topic and is worth a good read.
I saw the HNQ Why are quaternions more popular than tessarines despite being non-commutative? and wondered:
Quesiton: Are bicomplex numbers including tessarines ever used in spaceflight (as an alternative to quaternions)?
See Bicomplex number for more.