real and unreal

template<typename T> T              real(quaternion<T> const & q);
template<typename T> quaternion<T>  unreal(quaternion<T> const & q);

These return q.real() and q.unreal() respectively.

conj

template<typename T> quaternion<T>  conj(quaternion<T> const & q);

This returns the conjugate of the quaternion.

sup

template<typename T> T sup(quaternion<T> const & q);

This return the sup norm (the greatest among abs(q.R_component_1())...abs(q.R_component_4())) of the quaternion.

l1

template<typename T> T  l1(quaternion<T> const & q);

This return the l1 norm (abs(q.R_component_1())+...+abs(q.R_component_4())) of the quaternion.

abs

template<typename T> T  abs(quaternion<T> const & q);

This return the magnitude (Euclidean norm) of the quaternion.

norm

template<typename T> T  norm(quaternion<T>const  & q);

This return the (Cayley) norm of the quaternion. The term "norm" might be confusing, as most people associate it with the Euclidian norm (and quadratic functionals). For this version of (the mathematical objects known as) quaternions, the Euclidean norm (also known as magnitude) is the square root of the Cayley norm.