Given a star-algebra $(A ,\ast)$ in characteristic zero, then for $a\in A$ any element
its real part is $Re(a) \coloneqq \frac{1}{2}(a + a^\ast)$;
its imaginary part is $Im(a) \coloneqq \frac{1}{2}(a - a^\ast)$.
Standard situations where real and imaginary parts are considered include the real normed division algebra
$A$ the complex numbers;
$A$ the quaternions;
$A$ the octonions