The given complex $[Pt(NH_3)_4][CuCl_4]$ consists of a complex cation and a complex anion. Such complexes exhibit coordination isomerism, which arises due to the interchange of ligands between the cationic and anionic entities of the different metal ions. The possible coordination isomers (which are ionic in nature) are:

1. $[Pt(NH_3)_4]^{2+}[CuCl_4]^{2-}$
2. $[Pt(NH_3)_3Cl]^{+}[Cu(NH_3)Cl_3]^{-}$
3. $[Cu(NH_3)_4]^{2+}[PtCl_4]^{2-}$
4. $[Cu(NH_3)_3Cl]^{+}[Pt(NH_3)Cl_3]^{-}$

Note: The combination where two ligands are exchanged, i.e., $[Pt(NH_3)_2Cl_2]$ and $[Cu(NH_3)_2Cl_2]$, results in neutral molecules rather than a complex salt, and is therefore not counted as a coordination isomer of the given salt. Thus, there are 4 isomers.