Step 1: Understand homoleptic complex.
A homoleptic complex is a coordination compound in which the central metal atom is attached to only one type of ligand.
If more than one type of ligand is present, the complex is called heteroleptic.
Step 2: Analyze each option.
Option (1):
\[
[Co(NH_3)_4Br_2]^+
\]
This contains two types of ligands:
\[
NH_3 \text{ and } Br^-
\]
Hence, it is heteroleptic.
Option (2):
\[
[Co(C_2O_4)(NH_3)_4]^+
\]
This contains oxalate and ammonia ligands.
Hence, it is heteroleptic.
Option (3):
\[
[Co(NH_3)_6]^{3+}
\]
This contains only one type of ligand:
\[
NH_3
\]
Hence, it is a homoleptic complex.
Option (4):
\[
[Co(CN)_4Cl_2]^-
\]
This contains cyanide and chloride ligands.
Hence, it is heteroleptic.
Step 3: Final conclusion.
Therefore, the homoleptic complex is
\[
\boxed{[Co(NH_3)_6]^{3+}}
\]