We study the motion of a hole with internal degrees of freedom, introduced to the zigzag magnetic ground state of Na$_2$IrO$_3$, by using the self-consistent Born approximation. We find that the low, intermediate, and high-energy spectra are primarily attributed to the singlet, triplet, and quintet hole contributions, respectively. The spectral functions exhibit distinct features such as the electron-like dispersion of low-energy states near the $\Gamma$ point, the maximum M-point intensity of mid-energy states, and the hole-like dispersion of high-energy states. These features are robust and almost insensitive to the exchange model and Hund's coupling, and are in qualitative agreement with the angular-resolved photoemission spectra observed in Na$_2$IrO$_3$. Our results reveal that the interference between internal degrees of freedom in different sublattices plays an important role in inducing the complex dispersions.
