I often come across students who are confused by the idea that the complex unit, $i$, is defined as $i^2 = -1$. Since we are using the complex numbers in an engineering course, we use the complex numbers to encode information about a sinusoid of a given frequency, where the argument is the phase of the sinusoid with respect to a reference and the magnitude is the amplitude. I've started to explain the need for the complex numbers by re-defining the complex unit as: $i^5 = i$, which helps to explain that the complex numbers inherently encode periodic information, hence why they were so useful for encoding sinusoids.
My question is this: conceptually this helps the students to better understand why we use the complex numbers (they tend to hate the fact that we use the term "imaginary number"). However, this isn't the original definition of $i$. Is there anything seriously wrong with approaching the concepts with my definition? Is there anything special about the fact that $i^2 = -1$ that is not also present in my definition?