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The whale optimization algorithm (WOA) is inspired by humpback whales' bubble-net assaulting mechanism and imitates behaviors such as shrinking and enveloping prey, spiral bubble-net assaulting prey, and variational searching for prey to determine the best solution. However, the basic WOA has the disadvantages of low computation precision, slow convergence rate and easily falling into search stagnation. To strengthen the optimization quality and search reliability, this paper presents a distinctive complex-valued encoding WOA (CWOA) to satisfy the function optimization and engineering design. The complex-valued methodology utilizes a diploid structure to encode individual whales, and the real and imaginary components are added to the basic WOA to revise the position of each humpback whale, which converts the two-dimensional encoding area to a one-dimensional expression area and utilizes the real and imaginary components to illustrate a feasible solution with inherent parallelism. This methodology enriches the population diversity, furthers the individual information, elevates the general search ability, avoids premature convergence and promotes convergence efficiency. The CWOA not only utilizes the characteristics of the complex-valued methodology to avoid slipping into local optima but also exhibits excellent adaptability and robustness to determine the accurate value. Sixteen benchmark test functions and eight engineering designs are used to verify the applicability and practicability of the CWOA. The experimental results demonstrate that the optimization productivity and search performance of the CWOA are superior to those of other algorithms. In addition, the CWOA is a more successful and efficacious method that completely stabilizes exploration and exploitation to establish a quicker convergence rate, higher computation precision, greater resilience and stability. [ABSTRACT FROM AUTHOR]