Deactivation of excited electronic states through coupling to dissociative (1)pi sigma* states in heteroaromatic systems has received considerable attention in recent years, particularly as a mechanism that contributes to the ultraviolet (UV) photostability of numerous aromatic biomolecules and their chromophores. Recent studies have expanded upon this work to look at more complex species, which involves understanding competing dynamics on two different (1)pi sigma* potential energy surfaces (PESs) localized on different heteroatom hydride coordinates (O-H and N-H bonds) within the same molecule. In a similar spirit, the work presented here utilizes ultrafast time-resolved velocity map ion imaging to study competing dissociation pathways along (1)pi alpha* PESs in mequinol (p-methoxyphenol), localized at O-H and O-CH3 bonds yielding H atoms or CH3 radicals, respectively, over an excitation wavelength range of 298-238 nm and at 200 nm. H atom elimination is found to be operative via either tunneling under a (1)pi pi*/(1)pi sigma(O-H)* conical intersection (CI) (298 >= lambda >= 280 nm) or ultrafast internal conversion through appropriate CIs (lambda <= 245 nm), both of which provide mechanisms for coupling onto the dissociative (1)pi sigma(O-H)* state associated with the O-H bond. In the intermediate wavelength range of 280 >= lambda >= 245 nm, (1)pi sigma(O-H)* mediated H atom elimination is not observed. In contrast, we find that (1)pi sigma(O-CH3)* state driven CH3 radical elimination is only observed in the excitation range 264 >= lambda >= 238 nm. Interpretation of these experimental results is guided by: (i) high level complete active space with second order perturbation theory (CASPT2) calculations, which provide 1-D potential energy cuts of the ground and low lying singlet excited electronic states along the O-H and O-CH3 bond coordinates; and (ii) calculated excitation energies using CASPT2 and the equation-of-motion coupled cluster with singles and doubles excitations (EOM-CCSD) formalism. From these comprehensive studies, we find that the dynamics along the O-H coordinate generally mimic H atom elimination previously observed in phenol, whereas O-CH3 bond fission in mequinol appears to present notably different behavior to the CH3 elimination dynamics previously observed in anisole (methoxybenzene).