Although the name "imaginary numbers" may suggest some special issue about existence, I think the general view would be that the existence of so-called imaginary numbers is no more and no less problematic than the existence of more familiar numbers, including zero, negative numbers and irrational numbers, all of which were considered puzzling or problematic when they first entered mathematics. Numbers, if such there be, are abstract objects of a certain sort. Whether there really such things as abstract objects at all is something that philosophers have long argued about. A bit too crudely, Platonist say yes, and nominalists say no. So if there aren't any abstract objects, then i5, for example, doesn't exist, but then neither does 5. If there are abstract objects then there's no clear reason to worry about whether 5 exists. And since the extension of the real number system to the complex number system is mathematically straightforward, there would be no clear reason to let 5 in but keep i5 out.

Although the name "imaginary numbers" may suggest some special issue about existence, I think the general view would be that the existence of so-called imaginary numbers is no more and no less problematic than the existence of more familiar numbers, including zero, negative numbers and irrational numbers, all of which were considered puzzling or problematic when they first entered mathematics. Numbers, if such there be, are abstract objects of a certain sort. Whether there really such things as abstract objects at all is something that philosophers have long argued about. A bit too crudely, Platonist say yes, and nominalists say no. So if there aren't any abstract objects, then i5, for example, doesn't exist, but then neither does 5. If there are abstract objects then there's no clear reason to worry about whether 5 exists. And since the extension of the real number system to the complex number system is mathematically straightforward, there would be no clear reason to let 5 in but keep i5 out.