The conjugate of a complex number can be very useful when you need a real component;e.g. when rationalizing to remove the imaginary part from the denominator of a fraction.
Some useful properties:
If z is a complex number and z* is its conjugate (z* is used in some applied fields, while zbar ( a z with a bar over it) is used in most math texts)
z=a+bi, z*=a-bi where a,b are real numbers and i is the square root of negative 1.
z+z* is real.
(z)(z*) is real.