The only situation in which two angles are both congruent and complementary is when their measures are both 45°:
Two angles are complementary if their sum is 90°, and they are congruent if their measures are the same, say n. Therefore, to find two congruent angles which are complementary, we solve the following equation:
n+n=2n
2n = 90
2n/2 = 90/2
n=45