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I don't really understand. What does "3D analogue of the complex numbers" mean?
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the complex plane is partitioned by two axes, the real and the imaginary axis. In the quaternion space, it is partitioned by 4 axes, the i-axis, the j-axis, and the k-axis and of course the real usual real axis. The differences are the different ways of doing addition, subtraction, and multiplication. I am not sure whether quaternion can do division. Nevertheless, quaternion is really composed of a vector part and a scalar part. Later, Heaviside was able to separate them into just vectors and scalars.