Beyond S\'ersic + exponential disc morphologies in the Coma Cluster

[abridged] We explore the diversity of internal galaxy structures in the Coma cluster across a wide range of luminosities ($-17$\,$>$\,$M_g$\,$>$\,$-22$) and cluster-centric radii ($0$\,$<$\,$r_{\rm{cluster}}$\,$<$\,1.3 $r_{200}$) through analysis of deep Canada-France-Hawaii Telescope $i$ band imaging. We present 2D multi-component decomposition via GALFIT, encompassing a wide range of candidate model morphologies with up to three photometric components. Particular focus is placed on early-type galaxies with outer discs (i.e. S0s), and deviations from simple (`unbroken') exponential discs. Rigorous filtering ensures that each model component provides a statistically significant improvement to the goodness-of-fit. The majority of Coma cluster members in our sample (478 of 631) are reliably fitted by symmetric structural models. Of these, 134 ($28\%$) are single S\'{e}rsic objects, 143 ($30\%$) are well-described by 2 component structures, while 201 ($42\%$) require more complex models. Multi-component S\'{e}rsic galaxies resemble compact psuedobulges ($n\sim$\,2, $R_e \sim$\, 4 kpc) surrounded by extended Gaussian-like outer structures ($R_e > 10$ kpc). 11\% of galaxies ($N=52$) feature a break in their outer profiles, indicating `truncated' or `anti-truncated' discs. Beyond the break radius, truncated galaxies are structurally consistent with exponential discs, disfavouring physical truncation as their formation mechanism. Bulge luminosity in anti-truncated galaxies correlates strongly with galaxy luminosity, indicating a bulge-enhancing origin for these systems. Both types of broken disc are found overwhelmingly ($>70\%$) in `barred' galaxies, despite a low measured bar fraction for Coma ($20\pm2\%$). Thus, galaxy bars play an important role in formation of broken disc structures. No strong variation in galaxy structure is detected with projected cluster-centric radius.