[nilkn]

If by real plane you just mean the standard vector space R^2, the answer is algebraic rather than topological. R^2 as a vector space has an addition operation, but no multiplication operation. The complex plane is obtained by simply picking the appropriate multiplication operator.

In general, completing the rationals into the reals is more complex than constructing the complex plane from the real numbers. For the latter, you just need to adjoin a single element (sqrt(-1)), enforce existing arithmetic rules, and the rest falls into place. For the former, you can't just adjoin a single new element like sqrt(2). Doing so will get you the ring (actually field) Q[sqrt(2)], but not R.

If you take R and adjoin two special new elements (sqrt(-1) and the point at infinity), you do obtain a topologically different result: the Riemann sphere. This sphere is in many ways the more natural domain for complex analysis than the complex plane.