Can real numbers be dense in complex numbers for some topology? I am looking for a topology such that R is dense in C .
I was thinking I can construct a surjective continuous function f : R → C such that the image of Q is R.
 A: The Zariski topology is an important “real-world” topology having this property. In a complex affine space (maybe not just a complex line), the Zarisiki topology is the topology whose closed sets are the zeroes of polynomial functions. In dimension 1 the closed sets in the Zariski topology are just the finite subsets, together with the full complex line, which indeed makes the real line a dense subset.
The Unitarian trick is a great example of an argument involving distinct topologies on some space (in this case the Zariski topology and the Euclidean topology) to prove a meaningful theorem.
There is a lot of other topologies making the real line a dense subset of the complex plane, but only a few of them are interesting. Some already quoted the coarse topology. If you are interested in producing a lot of rather artificial examples, take any map $f$ from the complex plane onto a finite set $F$, so that the restriction of $f$ to the real line is also onto. The set of preimages through $f$ of the poert set of $F$ are the closed sets of a topology on the complex plane where the real line is dense.
A: There are many many possibilities; the following is probably the simplest. 
 Take the indiscrete topology on $\mathbb{C}$, in which the only open sets are $\mathbb{C}$ and $\emptyset$.  Then every nonempty subset of $\mathbb{C}$ is dense.
A: Take an automorphism of $\Bbb C$ other than the identity or complex conjugation. These exist by Zorn's lemma. The image of $\Bbb R$ under this
automorphism is an isomorphic copy of $\Bbb R$ dense in $\Bbb C$.
