The aim of thesis is to develop complex multiplication and exponentiation algorithms that compute results with low normwise and componentwise relative error. In the process, we develop a method of error analysis for complex functions. We correct complex multiplication efficiently using a technique for doubling the working precision and demonstrate that the algorithm is corrected. Next, we discover the underlying source of error in the standard algorithm for computing complex exponentiation, and show that the algorithm is not corrigible using working precision. A corrected algorithm using extended precision is developed, and an example implementation and testing are presented.