I was unsure about how to actually write the question.
I know that in the real number axis we have the naturals, integers, rationals, irrationals and reals; and in the imaginary number axis we have the imaginary numbers, together forming the complex number plane. I also know that adding dimensions to that plane we can have numbers of dimension $2^n$ (quaternions, octonions, etc).
What I want to know is: Are more numbers outside of that? Has a type of numbers ever been created from scratch just to satisfy or demonstrate a given property, even if they are not used outside of that? Are there numbers that are not numbers by the previous definition?
Any reference or links to lists of number types other than those I have mentioned would be greatly appreciated.