Indeed, developing new notation should be not only allowed, not only encouraged, but even required as an assignment at various points during mathematical training, at every age level. Also to be encouraged is conscientious analysis and critique of existing notations, algorithms, and explanations.

Society advances in two ways: in small incremental steps, and in large leaps. When we stick to existing notations and concepts we can make incremental improvements. But the large leaps often come from creating better notations and better conceptual understandings, which can turn substantial amounts of knowledge and logical thinking into obvious consequences of an organized and intuitive system.