Should doctors be multi tasking while within surgery?

Absolutely. He has to be monitoring important signs and understanding what the implication and ramifications are when those fundamental signs change beyond confident limits. Fortunately, the surgical squad does most of the actual monitoring, but the surgeon has to accord with it when those signs changeover.
What about his other patients? Suppose while he is doing an appendectomy on someone, one of his other patients, utter you, or your mom, goes code blue. Do you want him to NOT check his see?
NO! surgery is very complicated and they should stay to one item only.
SteveA8 have no idea what he is chitchat about.
I am an anesthesiologist, so I spend adjectives day next to surgeons.
For the most part, they concentrate on the surgery. Occasionally, they take phone calls or messages, and later they take a short break (if appropriate) and agreement with the issue. Usually near is conversation during surgery, but not always, and never if things aren't going economically. Most OR's have music playing, but not too loudly.
Vital signs are monitored by the anesthesiologist. The surgeon cannot do both the anesthesia and the surgery (unless it's a thoroughly minor case below light sedation, and even consequently, there is a nurse monitoring the VS) We communicate problems near the surgeon, just to hold on to them informed, and most of the time, they reciprocate. (It's important for us to know, for example, if the surgeon expects plentifully of bleeding).
Surgeons do not leave one lenient to attend to another. They finish the operation. If there is an emergency, someone else handle it.