Brain-inspired computing refers to computational models, methods, and systems, that are mainly inspired by the processing mode or structure of brain. A recent study proposed the concept of "neuromorphic completeness" and the corresponding system hierarchy, which is helpful to determine the capability boundary of brain-inspired computing system and to judge whether hardware and software of brain-inspired computing are compatible with each other. As a position paper, this article analyzes the existing brain-inspired chips’€™ design characteristics and the current so-called "general purpose" application development frameworks for brain-inspired computing, as well as introduces the background and the potential of this proposal. Further, some key features of this concept are presented through the comparison with the Turing completeness and approximate computation, and the analyses of the relationship with "general-purpose" brain-inspired computing systems (it means that computing systems can support all computable applications). In the end, a promising technical approach to realize such computing systems is introduced, as well as the on-going research and the work foundation. We believe that this work is conducive to the design of extensible neuromorphic complete hardware-primitives and the corresponding chips. On this basis, it is expected to gradually realize "general purpose" brain-inspired computing system, in order to take into account the functionality completeness and application efficiency.