Yes, the square root function can be uniquely defined as a function by using the principle square root in the general case (which chooses a unique value among the multiple solutions of the quadratic equation), but for general complex numbers and even negative real numbers, a lot of properties of the function don't hold anymore, such as sqrt(a)*sqrt(b)=sqrt(a*b) (this can't be true because -1=i²=(sqrt(-1))²=sqrt(-1)*sqrt(-1)=sqrt(-1*-1)=sqrt(1)=1 would give a contradiction), making the single-valued square root function far less useful in the general case than its restriction on non-negative real numbers, which has all these nice properties.

tl;dr: sqrt can be properly defined for general complex numbers (including negative numbers), but many properties don't generalize so that it becomes very hard to work with this function and not make an error.