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A complex number is just two real numbers stitched together. It's used in many areas, such as the Fourier transform which is common in computer science is often represented with complex numbers because it deals with waves and waves are two-dimensional, and so rather than needing two different equations you can represent it with a single equation where the two-dimensional behavior occurs on the complex-plane.
In principle you can always just split a complex number into two real numbers and carry on the calculation that way. In fact, if we couldn't, then no one would use complex numbers, because computers can't process imaginary numbers directly. Every computer program that deals with complex numbers, behind the scenes, is decomposing it into two real-valued floating point numbers.
I don't think this is really an accurate way of thinking about them. Yes, they can be mapped to a 2d plane, so you can represent them with their two real-numbered coordinates along the real and imaginary axes, but certain operations with them (eg. multiplication) can be done easily with complex numbers but are not obvious how to carry out with just grid points. (3,4) * (5,6) isn't well-defined, but (3+4i) * (5+6i) is.