I'm going back to math as promised. Let's think about how and why the different kinds of numbers were created.

First there are "natural numbers". It's a concept that is easy to understand, because it can be used to count physical objects we see around us. We can also make some simple operations on them. We can add them - no problems here, because if we add two natural numbers, the result is also a natural number. But we can also subtract them, and that's where a problem arises: if we subtract a number that is bigger than the one we subtract from, the result is NOT a natural number. So what can we do? We can invent a new kind of numbers, called negative numbers, that are the results of such operations.

The same goes with rational numbers - they were created as the extension of the set of numbers in case division doesn't give us a whole number. Irrational numbers are used to solve some equations, as well as geometrical problems, like calculating the diagonal length in a square. There are also transcendental numbers, that appear in other kinds of operations, like calculating the length of a circle.

So does this process ever end? Yes, it does. The end of it are the complex numbers. They allow all kinds of operations on all kinds of operands. We can take a square root of a negative number, a logarithm of a negative number, and so on. Almost everything is possible. (Things like dividing by zero are still impossible, but that's another story.)

Some results of these operations are multi-valued. But is this something that only the complex numbers can do? For example, everyone (with basic math knowledge) knows that the square root of 4 is 2, because 2 squared is 4. But -2 squared is also 4, so it should be another value of the square root of 4. It's just a convention that we take only the positive value, and not the negative one. So operations with multi-valued results are not exclusive to complex numbers. Another thing is that some of these operations are quite complicated (look for the formula for a+bi to the power of c+di, it's total mess).

There are further extensions to the set of numbers, like quaternions, but they are more like artificial constructs. They are not needed to make any operations possible, they are just created to serve some purpose (quaternions are used to represent rotations in 3D space, for example). So complex numbers are the end of the process of making the set of numbers complete. And they appear in many areas of physics, so it seems nature uses them a lot. So I think they are the ones that truly deserve to be called "natural" numbers, and using names like "real" and "imaginary" (for the two parts of a complex number) is just wrong, and is the result of superstitions from the time they were introduced.