Nicholas,
Most of the objections you are raising are also applicable in simpler contexts. Do you believe that 
exists? The ancient Greeks knew and loved the integers and the rational numbers (ratios of integers), and they believed that these were the only numbers that "existed." That is, however, until was discovered. The ancient Greeks' love of geometry meant that they discovered in that context, specifically: what is the length of the diagonal of a square whose sides have length 1? When was proved to exist, and, furthermore to be a number that could not be written as a ratio of integers , the ancient Greeks were not particularly inclined to believe in it.
Using just and the rational numbers, you can construct a set of numbers that is "bigger" and more general than just the rationals. If you take a linear combination of a rational number and , you create a number (I had thought they were called "Euler numbers," but apparently my memory is wrong, according to Wikipedia. Anybody know what they're really called?) that behaves almost identically to the complex numbers; consider
where 
and 
are rational numbers. This new number 
is definitely not a rational number, and all you can "do" with it is to "push around" the and construct other numbers of this form. Despite this, this class of numbers is a valid generalization of the rational numbers, and it is one step toward constructing the entire spectrum of real numbers. In fact, for precisely this reason, the algebraic properties of this class of numbers are almost identical to the properties of the complex plane. So, truly, all your objections to the use of a "false" number like 
also apply to numbers that were previously thought to be false, such as .
On a more fundamental/philosophical level, what does it really mean for a number to "exist"? Does the number 1 exist? What is "one-ness"? What, precisely, is the trait shared in common between one desk, one bird, one tree, one person, one stone? Numbers are really an incredibly abstract concept that only tangentially has anything to do with the real (physical) world. Is it really any stranger to say that exists than that exists? Or 
? Or numbers even more complicated, such as quaternions?
Perhaps the truly miraculous and humbling thing about mathematics is that such purely abstract concepts as numbers have anything to do with our world at all. But yet they have given us the tools to describe our world at scales too small or to large or too powerful for us to experience in our day to day lives.
