Seeing the whole number numbers as two directions and comparing them with the complex numbers as a to dimensional object is a good idea.
As set the whole number are not really special, because the whole number and natural numbers are both countable sets.
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I am trying to point out the following:
So imaginary numbers is the set given by all the real numbers times i. So it is a continuous spectrum. Then it makes sense to compare them to the real numbers.
Similarly, when dealing with the integers in this setting you might want to compare them to the set given by multiplying all the integers times i.