The conjugate of a complex number a + bi is a - bi. There are various ways of denoting this, but the modern standard is, for the number A, A-bar.

The conjugate of a real number is, therefore, itself, hence this isn't often seen. The conjugate of a pure imaginary number is it's own negative - again, fairly useless.

To generalize the first definition to quaternions, the conjugate of a + bi + cj + dk is a - bi - cj - dk. This is frequently encountered in the multiplicate inverse:
a-1 = ~a / (a · ~a)
Where ~a is the conjugate of a.