I noticed that for each basic increasing binary function (addition, multiplication, and exponentiation) its inverse (or just a inverse) of certain values adds more number types to the number line (or plane):
subtraction --> all integers
division --> rationals
roots --> complex numbers
so can this be extended even further onto the inverse tetration, making a new number group, or can the complex numbers accommodate this? Furthermore would these numbers break normal algebraic rules such as commutativity