Yes. A complex number can be represented by z=a+ib where a and b are real. The real and imaginary components remain separate but this does not prevent complex numbers from being added, subtracted, multiplied or divided. Certain operations (e.g., multiplication and division, a+ib multiplied by a-ib gives a^2+b^2 which is totally real) can actually transform the imaginary components into real components. The results of operations always produce results which have a real and imaginary part (including zero real or imaginary). There is no sense of "more complex" introducing another "dimension"; it is only necessary to have just two components, real and imaginary.
Adding two complex numbers means you add the real and imaginary parts separately: example:
1+2i + 3-4i=4-2i. The two components are separately preserved.