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You can construct very general quaternion algebras in Sage, and the command QuaternionAlgebra(QQ, 2, 3) would construct an algebra with base field the rationals, with elements i and j such that i^2=2 and j^2=3. You've constructed a quaternion algebra over the "symbolic ring" SR with elements i and j such that i^2 = j^2 = -1.

Maybe because they are constructible in such generality, exponentials are not implemented for them. You can just use the formula on the wikipedia page (https://en.wikipedia.org/wiki/Quaternion#Exponential,_logarithm,_and_power_functions). General quaternion algebras have their own wikipedia page, too: https://en.wikipedia.org/wiki/Quaternion_algebra.