Changing a broken lightbulb when you can't unscrew the base presents obvious problems. Try using a cut potato. It's important to first unplug the lamp. Carefully remove any large pieces of glass that stick out, using a triple thickness of paper towels and working above the trash can. Then, gently press the cut side of the potato into the remnants of the broken bulb; once potato is secure, slowly twist to unscrew the bulb.
Removing a Broken Lightbulb
Organizing Good Things 2005, May 2005