“Air purifiers” commonly refer to some sort of particulate filter.
DANGER
PARTICULATE FILTRATION DOES NOTHING FOR VOCs.
“But my filter has carbon in it too”: Common residential air purifiers do not have enough carbon fiber, dwell time, or clean air exchange rate to handle VOCs like styrene from your printer.
Most air filters with carbon have maybe enough to filter the smells from a fart (that’s what they’re designed for).
The amount and type of VOCs emitted by FDM can’t be negated by most residential or prosumer filtration systems.
You should, of course, do your own research and make a fact-based decision based on your risk tolerance.