Are there ever any complications for going under anesthesia?

YES. Similar to undergoing surgery, there are risks associated with the procedure you are having. Anytime you go under anesthesia, there are risks with that as well. You could have an allergic reaction to the medication given IV or that you breath during surgery. Before any procedure or anesthesia, your doctor should talk to you about the risks.
Minimal. There are some risks associated with general anesthesia, but they are minimal. The risk of dying under anesthesia is about one in 350, 000. It is much more likely that a patient will die driving to the hospital for their surgery. Several other risks, including nausea and vomiting and poor pain control, are also concerns, but they can be managed by a trained anesthesiologist.
They are rare. Very unlikely. Allergic reactions are rare and can be treated. Other anesthesia complications include malignant hyperthermia and pseudocholinesterase related issues. Both of these are rare.