The aim of this paper is to introduce for the first time the basic concept of the first-order Weak Fuzzy Complex- Initial Value Problems (WFC-IVPs). We find, using a special isomorphic transformation function, that solving a WFC-IVP is equivalent to solving two classical real IVPs with respect to their own real variables. Thus, we study “the existence and uniqueness”, “the stability”, and “the well-posedness” associated with the WFC-IVPs in terms of definitions, lemmas, and theorems. Then, we get the approximate solutions of a WFC-IVP by stable and convergent numerical methods. One of the most famous and simple methods to solve IVPs is Euler’s method, which is discussed for well-posed WFC-IVPs. However, we focus on a stable linear model WFC-IVP with real coefficients and real initial values, and we further investigate the properties of the results. Additionally, we present an example with tables and diagrams of its numerical solutions and absolute errors by Python to clarify how Euler’s algorithm works.