Mondor's Cord, also known as "superficial thrombophlebitis" is not so much a complication as it is a minor annoyance. Basically, Mondor's Cord is nothing more than inflammation of the veins. It is usually caused by trauma (surgery in the case of breast augmentation). Doctors usually recommend warm compresses and ibuprofen for treatment, although many women manage fine without these things. While Mondor's Cord can cause some discomfort, it is by no means "painful", nor does it develop into a more serious complication. It resolves by itself in a matter of days to weeks.
Mondor's Cord, pictured above, may appear in the crease of the breast, or in the armpit / upper arm, although the latter is usually only seen in women that have transaxillary breast augmentation. Additionally, there may only be inflammation of one vein, or there may be several that inflamed and swollen. It is never anything to be alarmed about, but always let your surgeon know about things like this, just so that he or she can be sure to add it to your medical records.