No more so than integers or differential equations.
Complex (real and imaginary joined) numbers have several useful properties. For an example, if one solves x^2=2 (going back to Pythagoras), one finds that there is no rational p/q equal to x, so x is of a different kind than fractions. However, x^2=3 needs a new number, just attaching x from x^2=2 doesn't help solve x^2=3. Attaching all the (infinitely many) x^2=p for all prime integers p, doesn't solve the system x^3=2. It gets worse, attaching all the roots of all the integers gives neither pi, nor e. Eudoxus (and Dedekind and Cantor) gave complete discriptions of the real numbers, but these descriptions are rather long and technical.
With imaginary numbers, things are a bit different. One introduces the solution to x^2+1=0, (named i for this note.) Then the solutions to x^3+1=0, x^245+24x^44=0, and all polynomials in x have a solution involving i. Even equations with complex coefficients such as (3x+2i)x^5+7=0 have solutions in complex numbers. This is the Fundamental Theorem of Algebra.
Complex numbers are not just pairs of reals. They are pairs of reals with special rules for arithmetic. They form a field. The rules are (a+bi)+(c+di)=(a+b+(c+d)i) and (a+bi)*(c+di)=(ab-cd+(ad+bc)i). One gets the same result by just using i^2=1 and treating i as a "number."
These complex numbers describe electrical and magnetic fields and are useful in Fourier series because of the identity e^(ix)=Cos(x)+i*Sin(x) (Euler's relation.) Quantum mechanics requires complex numbers for its descriptions. For example, probabilities occur in QM as squares of numbers (actually squares of averages over wave functions) but the numbers are complex (the "square" of a complex number (a+bi) is actually taken as (a+bi)*(a-bi)= a^2+b^2, a-bi is called the conjugate of a+bi.) When two "waves" add in quantum mechanics, the complex numbers describing are added and the square of the result taken to give probabilities. (Classically, one would just add the probabilities.)
Complex numbers are essential in analyis. For example, take a simple function f(x)=1/(1+x^2) with real x. A power series for this function will only converge if |x| < 1. This is because there is a zero at x=i (or x=-1.) Things happening in the complex numbers are intruding onto the real line.
Imaginary numbers are just as "real" (in the ontolgical sense) as real numbers (or just as imaginary.) The terms are unfortunate as ordinary language uses the words "real" and "imaginary" differently. (Not surprisingly terms like: group, field, radical, or, implies, if, then, else, not, and, root, function, etc. have technical meanings that don't correlate well with ordinary usage.)
Ha! I guess there that's where "the uncertainty principle" basically comes from... which presumably can only be "tamed" by stochastic methods?
Somehow I suspect there's an analogy between pure mathematics and the way natural phenomena unfold and evolve in your observation. Thank you so very much, Doc, for your analysis. I will definitely study it further. (I mean that.)