What is the most commonly used metric in $\mathbb{C} \times \mathbb{C}$? I am reading Conway's Functions of One Complex Variable, and many times it makes reference to a function from $\mathbb{C} \times \mathbb{C}$ being continuous. I would like to rigorously prove such assertions, so I have to know which metric he is referring to. I haven't taken topology, so I'm not sure what the most natural or nice metric would be. The one I've come up with is $d[(z_1, w_1), (z_2, w_2)] = |z_1-z_2| + |w_1 - w_2|$.
Is there a "natural" metric that is most often used on $\mathbb{C} \times \mathbb{C}?$ 
 A: The concept of a continuous map between metric spaces is usually introduced via the $\varepsilon$-$\delta$-definition. In a more general form your question is this: If we are given metric spaces $(X_1,d_1), (X_2,d_2)$, how do we define a metric $\Delta$ on $X_1 \times X_2$?
There is no standard definition of $d$, but a standard requirement. Here it is:
For any metric space $(Y,d)$, a map $f : (Y,d) \to (X_1 \times X_2,\Delta)$ is continuous if and only if the two maps $p_i \circ f : (Y,d) \to (X_i,d_i)$, where $p_i : X_1 \times X_2 \to X_i, p_i(x_1,x_2)= x_i,$ are continuous. If you take a topology course, this requirement will become very clear. At the moment accept it as it is.
You will see that there are many solutions for $\Delta$. In your answer you gave one, and in the comment by ZeroXLR you have another one. However, all metrics with the desired property turn out to be equivalent, and this means that it is not really important which you choose. In topological terms the essential point is that the product of topological spaces can be endowed with a unique standard topology with reasonable properties.
