Question: what is the absolute value of a quaternion?
Answer: The absolute value of the quaternion q = a + b*I + c*J + d*K is abs(q) = sqrt( a^2 + b^2 + c^2 + d^2 ). The real part of the quaternion q, like the definition of the real part of a complex number, is Re(q) = a. The imaginary part is defined as the complement to the real part: Im(q) = b*I + c*J + d*K.