A homonuclear complex has all coordinating atoms of the same element. Let's check:
- (1) \([\text{Co}(\text{NH}_3)_4\text{Br}_2]^+\): Coordinating atoms are N (from \(\text{NH}_3\)) and Br. Not homonuclear.
- (2) \([\text{Co}(\text{C}_2\text{O}_4)(\text{NH}_3)_4]^+\): Coordinating atoms are O (from oxalate) and N. Not homonuclear.
- (3) \([\text{Co}(\text{NH}_3)_6]^{3+}\): All coordinating atoms are N (from \(\text{NH}_3\)). Homonuclear.
- (4) \([\text{Co}(\text{CN})_4\text{Cl}_2]^0\): Coordinating atoms are C (from \(\text{CN}^-\)) and Cl. Not homonuclear.
Thus, the homonuclear complex is \([\text{Co}(\text{NH}_3)_6]^{3+}\).
Thus, the correct answer is option 3.