Would there be a purpose for a set of compressors in an electric turbo fan? This is actually a two part question. Would there be a purpose to a compressor in an electric turbofan where there is no need for a combustion chamber, and if not, is there then a purpose for a bypass as well?

Sorry if I'm unclear, as i am not an industry professional perhaps my wording may not be clear. What I mean to say is that in a regular turbofan engine there are compressors that push air into a combustion chamber with higher density (presumably to get more potential energy per volume of air) where fuel is introduced and ignited to create higher pressure/kinetic energy to run the turbines,  which in turn move the front compressors and the fan. However, in an electrically driven fan, since there is no combustion happening, and the fan is driven by an electric motor, does the engine then need the compressors? Do they perform any benefit in a system that does not use combustion to create pressure?

Then if not, is there any benefit to the bypass system in an electrically driven fan engine? Or does it make more sense to simply switch to a ducted fan system?
No.  The only purpose of the compressor is to provide high pressure air for the combustion chamber and turbine.  If there is no combustion chamber and no turbine, then you do not need a compressor.  Just drive a big ducted fan (or propellor, your choice) with your electric motor.