What are the complications I should take note of after jaw surgery?

In order to decide which method would most effective, we would need to assess and treat the cause of the protrusion. If it is due to protruding upper teeth which are flaring out and causing the lip to come forward, braces or Invisalign treatment would be ideal to reduce the protrusion. There may be a possibility that extractions of some teeth may be required to provide some space for the upper front teeth to move back into. Sometimes, it may be due to the imbalance of the upper and lower jaws.

It is great that you are thinking about correcting your bite. With skeletal issues such as a slanted jaw, surgery is indeed required in order to achieve a good stable bite. However depending on the degree of skeletal discrepancy (slanted jaw) braces / invisalign are still options for you to consider in order to straighten out your teeth and give you a good smile. Straight and aligned teeth can be achieved but there are limitations so it depends on what your ideal smile is! My suggestion would be to consider all your options and if needed seek a second opinion.