Other complex systems

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My question would be very short. As we all know, there are complex, quaternion number systems, which are based on multiplication and roots. So, my question is... Is there any other complex number systems that are NOT based on number roots? For example, to assume that sin(x) = 2 has some solution, it is not something different than imaginary (complex) number, because circle's equation shows it. So, does other complex number systems exist?