I owned a 06 BMW 750I with Bridgestone 245/50R/18 run flat tires on. I would like to know if I choose to put non run flat tires on would my tire pressure monitoring system
Still work? Does anybody have any recommendation on another name brand tire because
Bridgestone Turanza EL42 does not have a good mileage life, or should I just replace them
With the same kind? Please help. Thank you in advance.

I would recommend staying with your vehicles original tires or another industry compatible tire. High mileage tires don't often have the same handling and braking characteristics, so it's a compromise between handling or mileage.
As for the monitors, they monitor pressure with in the tire and wheel assembly, not the actual tire. After tire replacement, they may need recalibration in order to operate properly.
Some tire shops may be knowedgable in this area but I would check your owner's manual or call your dealer for advise.

Not all systems monitor actual pressure. Some use the relative speeds of the rotation of the wheels.

They usually need to be reset somehow when they indicate either through a scan tool or some sequence of events.

Others, have a specific deflate, inflate tire sequence that needs to be followed:

Say something like: Deflate all tires below x psi. Starting at the right rear and continuing counterclockwise inflate to the correct pressure.