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What, specifically, is the advantage of using the field of complex numbers over simply R^2 (the real plane).

I know that neater/less notation may sometimes help, but this would hardly be motivating to pursue the huge area of complex numbers.

I also know that holomorphic functions have very nice properties such as ease of finding the integral around a loop and that holomorphic functions satisfy the Laplace equation. I can see that this will be useful in physics for when all your functions satisfy the Laplace equation.

Basically, are these the only things which make the complex numbers more desirable than just using the real plane? Examples of such things would be nice (but be gentle, I'm a mathematician, not a physicist ;D). Also, aren't they a bit restrictive in the sense that they are only 2-dimensional... can they be used for example in steady fluid flow for 3-dimensions which would be much more complete than having only 2?

I'm also interested in looking at the mathematics of string theory; why do complex numbers have importance here?