Let \(\mathbb{C}\) be the set of all complex numbers. We define a partial order \(\precsim\) on \(\mathbb{C}\) as follows. For two elements \(z_1,z_2\in\mathbb{C}\), \(z_1\precsim z_2\) if and only if \(\text{Re}(z_1)\leq\text{Re}(z_2)\) and \(\text{Im}(z_1)\leq\text{Im}(z_2)\). By using this notion, the concept of complex-valued metric spaces is defined, that is, an ordered pair \((X,d)\) is a complex metric space if \(X\) is a nonempty set and \(d:X\times X\to\mathbb{C}\) is a function satisfied the following conditions: (a) \(d(x,y)\precsim0\) for all \(x,y\in X\); (b) \(d(x,y)=0\) if and only if \(x=y\); (c) \(d(x,y)=d(y,x)\) for all \(x,y\in X\); (d) \(d(x,y)\precsim d(x,z)+d(z,y)\) for all \(x,y\in X\). Moreover, the notions of convergent sequences, Cauchy sequences, and completeness are given. The authors of this paper continue the study of fixed point theorems in this setting. Some examples are also discussed.