Code

Is there anybody from Chicago that can tell me if it's ok to run 4" PVC underground for a bathroom?

Residential, commercial, inside, outside, under slab, crawl space? Not from that neck of the woods but around here anything goes when it's run to a septic tank. On sewers ductile iron is required from the building to the public sewer line. Local codes vary.

In Chicago you can use PVC for DWV in residential applications for 3 stories and under. That's aboveground. As far as underground goes, I don't know, but would guess the answer is no. Be aware, if this is permitted job, the inspector will tell you the answer. If you don't pull a permit and get caught, you are in big trouble. Unless you are an off duty fireman.