A complex number may be expressed as the sum of a real and imaginary part as follows:
a + i b
A quaternion adds two additional and independent imaginary parts as follows:
a + i b + j c + k d
So this adds two extra dimensions which square to a negative number, giving a total of:
- One dimension which squares to a positive number (real part)
- Three dimensions which square to a negative number (3 imaginary parts)
It does superficially look like quaternions extend the way that complex numbers represent rotations, but I don't think quaternion rotation is an extension the way complex numbers represent rotations, they are completely different. I think it is just a coincidence that they both happen to represent rotations. (if it is valid to use the word 'coincidence' in mathematics). For instance:
- The two dimensions in complex numbers (real and imaginary) can represent coordinates of the objects being rotated. The four dimensions of quaternions have no direct relationship to the 3 dimensions of the objects being rotated.
- In complex numbers rotation is done using complex exponent, in quaternions its done using the 'sandwich' multiplication.
- In complex numbers 'i' represents 90 degree rotation, in quaternions 'i' represents 180 degree rotation.