2008 328Xi BMW engine bolt broken. Tips for extracting the remains.

Is that your intake manifold? If so remove the manifold to expose the bolt and remove. Or whatever that bolt was holding on. If there is access
 
Yes, designed to break and allow the engine to drop and slide under car instead of crushing the occupants in front seats in head on collision.

4 bolts total.

This one was damaged when I drove over a pothole.

Its a known issue with BMW
 
Aluminum bolt, intake manifold??? Is there a gas leak on here? Engine Mount next to the oil pan it looks like. Put the craddle.on the support the engine and loosen the other bolts that mount it to the block. Try the left handed bit. If that doesn't work I would get the mount out of there and try to weld a nut to it. If that doesn't work you're gonna have to center punch it and drill it. Don't break an extraction tool.off in it. What ever you do. Take your time to get it centered. Don't drill too deep.
 
Looks like it isn’t broken flush with the block so you should be able to get some vice grips on it. Support the engine either from above or below. Remove the engine mount arm to get the best access. Apply heat if it doesn’t come loose right away
 
How much room do you have in there? Thought about drowning it in PB Blaster, torching it as hot as possible, then using an extractor bit?
 
I think The D has it right. I've done one or two (Mercedes and BMW) where the owner rode around on collapsed engine mounts until the vibration broke the bolts. If there's enough to grab, they usually spin right out. If it's recessed a little, you may be able to get a pick in there and start it moving. Any way you do it, taking the mount arm off will give you more room and less cursing.
 
Way to much work -
Option 1- raise the lift to highest position and push it off. Gotta post pictures!
Option 2 - trade it in.

Hoping to see pics in the near future.
 
This is exactly my wheelhouse(mostly Merc for ~13yrs). If you can get it to turn using an angle drill and left hand bit and extractor great, do that. Anything more than that you need to take the mount and arm off to get as much room to work as possible. Don’t forget to disconnect the battery if the alternator is right there
 
Get the next size larger bolt cut length to just protrude from hole.
Machine the OD of the bolt so that it fits the clearance hole.
Drill through the center of the bolt leaving maybe a 2mm wall.
Stick in hole, weld with mig (aluminum) welder, remove bolt.