How can I find an isomorphism between $\dfrac{\mathbb C}{L}$ and $\dfrac{\mathbb R^2}{\mathbb Z^2}$ where $\{w_1,w_2\}$ is a basis for $\mathbb C$ over $\mathbb R$ and $L=w_1\mathbb Z\oplus w_2\mathbb Z$?
Is it true that
$$\frac{\mathbb C}{w_1\mathbb Z\oplus w_2\mathbb Z}\simeq\frac{w_1\mathbb R\oplus w_2\mathbb R}{w_1\mathbb Z\oplus w_2\mathbb Z}\simeq\frac{w_1\mathbb R}{w_1\mathbb Z}\oplus\frac{w_2\mathbb R}{w_2\mathbb Z}\simeq\frac{\mathbb R}{\mathbb Z}\oplus \frac{\mathbb R}{\mathbb Z}\simeq\frac{\mathbb R^2}{\mathbb Z^2}\ ?$$
If yes, under which condition does this hold ?
Yes. All these isomorphism are canonical, so no condition is needed on it. Do this for any groups $M,M',N,N'$ where $M'$ and $N'$ are subgroups of $M$ and $N$, respectively (this holds for an arbitrary number of groups and the proof is analogous).