I remember going down that path when I was an undergraduate. Fortunately I had a very experienced theorist to hand who explained that, yes, people did try that, but stopped bothering because it doesn't generalize to general relativity, and there was no point keeping two mathematical toolboxes around when you could have one.
Clifford algebras are great for both SR and GR and have all the properties that one likes in quaternions (and more)
There are equivalent formulations of SR using quaternions.
But do they generalize to GR?
No, which is why the quaternionic approach isn't generally presented in textbooks. I think taylor and wheeler's SR book touches on it as an aside.