Well, to be precise its for the
mathematical analysis that you use complex numbers. There are obviously ways to analyze without using complex numbers, but they're tedious. As n0_3sc has pointed out, Euler's Theorem expresses a direct mapping between complex exponentials and sinusoids. From that property follow some other properties of complex exponentials which form the basis for phasor analysis, frequency domain computation and (therefore) the mathematics of control system analysis. Fundamentally, Euler's Theorem is an important connection between one of the most important pair of signals in electrical engineering and complex numbers.
Another reason you will like to use the complex exponential in expressing solutions to linear differential equations is its relation to eigenvalue problems. Specifically,
D_{x}(e^{j\omega x}) = je^{jx}
D_{x}^{2}(e^{j\omega x}) = -\omega^{2}e^{jx}
Sure enough you could do this with sinusoids too (in fact that's what you're doing right now), but doing it with complex exponentials is a whole lot neater. Quite a few problems of interest in the time domain involve the solution of eigenvalue equations, where these properties are often used.
Finally, even though all this is sufficient reason to use complex exponentials and more generally complex numbers, both the Laplace and Fourier Transforms have kernels that are complex exponentials and are members of a more general class of transforms whose variants are heavily used in signal processing, control systems and lot of other allied areas of electrical engineering and mathematics.
And yes, you encounter j in EE rather than i, because historically i(t) has been used to denote "instantaneous" current, as Varun points out. (
http://www.perl.com/doc/manual/html/lib/Math/Complex.html differs...apparently, i has also been used to denote intensity
)