First and foremost there could be an allergic reaction to the anesthetic. General is considered to be more risky yet any anesthetic could bring on a negative reaction. Although extremely rare, it is possible to bleed post-operatively resulting in another surgery to control and drain the collected blood. Another possibility is hematoma (a collection of clotted blood), seroma (a collection of the watery portion of the blood) and thrombosis (abnormal clotting).
Loss of sensitivity is common, although temporary. Permanent sensation loss in the areola (nipple) area or breasts, in general, can and may happen. There is also a risk of excessive scarring or inner scar tissue. Also, you must have more views (films) taken when having a mammogram if you have breast implants -- especially overs.
There is also a risk of calcifications -- especially when there is a definite, thick capsule around the implant. And galactorrhea, which is when you start producing breast milk, is also a complication. This is usually remedied on its own and may stop spontaneously although some cases may need medication or implant removal. Although very rare, it is worth mentioning, full disclosure is the key to an informed consent.
Breast tissue atrophy (loss, shrinking) is a possibility. According to the FDA, "the pressure of the breast implant may cause the breast tissue to thin and shrink. This can occur while implants are still in place or following implant removal without replacement".
Necrosis (death) of the breast tissue, breast envelope and or incision line can happen. Although extremely rare. The chances of necrosis are increased after radioactive/chemotherapy treatment, if you smoke and have poor circulation, or have temperatetherapy or cryotherapy post-operatively.
Extrusion is also an extremely rare occurrence but a scary possibility. Extrusion of the implant is where your body rejects the prosthesis and pushes it out of the skin, like when a piercing is pushed out or like when a thorn or splinter is pushed from the body. Then the implant may become visible under the skin and must be removed before it breaks through resulting in possibly an infection and definite major scarring.
Infection: You could develop a post-operative infection and need to have the implant removed, the infection dealt with and still have to wait for several months before an additional surgery can be performed to re-implant. Infections usually occur with the first 4 to 6 weeks. Some possible infections and a more common one being Staphylococcus, or simply Staph.