Are you a die-hard Platonist? To predicate existence is otherwise than simply saying numbers are real. Kant pointed out that the predication of existence to the concept of number adds nothing. This is why we can speak of mental concepts as being existentially neutral.
In science, those things which are modeled with imaginary numbers (such as wave functions in quantum mechanics or time-like intervals in relativity) are things which cannot be proven to exist through direct experiment. In basic alternating current electronics, imaginary number equations of the type 'e' to some imaginary number x t power are used NOT in that way, but as an expedient approach to differentiating difficult sin and cos equations. Imaginary numbers in that case do not model electrical phenomena (because only the real part of those equations apply to actual electrical phenomena). If you don't believe me, take out your Feynman lectures. There is a fundamental difference between the two. My point again is that in both quantum mechanics and relativity (the two theories which provide the most explanatory power to date in science), imaginary numbers are used to model the most basic parts of these theories. And in each case, no direct scientific or experimental contact with those parts of the models can be made. Again, basic reality, as it is currently modeled, is actually NOT something that can be touched on by science. (Only inferred.) Surprising, isn't it? And my other point was that religious belief is similar in that way.