Mathematicians first approached complex numbers cautiously. Although it was clear that they were useful in solving certain problems at least formally (for example, they are needed to even make sense of the formulas we found in the previous lectures) what was not clear was that they made sense. Perhaps indiscriminate use of them would lead to contradictions.
Gauß solved this problem by realizing that one can define and its operations in terms of and its operations. As long as we are willing to accept that makes sense, then no contradictions will come up from the use of complex numbers.