The concept of additional spatial dimensions is a topic of speculation in some theories, such as certain versions of string theory. However, it's important to note that these theories are still highly theoretical and have not been experimentally confirmed.
In the context of the Higgs mechanism, which is a fundamental part of the Standard Model of particle physics, the number of spatial dimensions does not directly affect the masses of particles arising from the Higgs mechanism. The Higgs mechanism provides a mechanism for particles to acquire mass through interactions with the Higgs field.
The masses of particles in the Standard Model, including elementary particles like quarks and leptons, are determined by their coupling to the Higgs field. The strength of this coupling determines the mass of the particle, regardless of the number of spatial dimensions considered. The Higgs field, and the associated Higgs boson, interacts with particles in all three spatial dimensions that we are familiar with, and its effect on particle masses is well-described within this three-dimensional framework.
In theories with extra spatial dimensions, the specific details of the Higgs mechanism and its interactions may be modified, but it is difficult to make general statements without a specific theoretical framework in mind. The effects of extra dimensions on particle masses and interactions would depend on the specific properties and dynamics of those dimensions, which are not yet fully understood or experimentally validated.