As we know, we usually use the metric $d(x,y)=\sqrt{\displaystyle\sum_{i=1}^nd_{i}^2}=\sqrt{\displaystyle\sum_{i=1}^n(x_{i}-y_{i})^2}$ for $\mathbb{R}^n$.
We also use $d(x,y)=|x-y|$ for $\mathbb{C}$, but what is the natural way to define the metric for $\mathbb{C}^2$?
I read from books that there are several ways to define a product metrics, for example, if $M=X\times Y$, we can define$$d_{E}(p,p')=\sqrt{d_{X}(x,x')^2+d_{Y}(y,y')^2}$$ or$$d_{max}(p,p')=\max\{d_{X}(x,x'),d_{Y}(y,y')\}$$ or$$d_{sum}(p,p')=d_{X}(x,x')+d_{Y}(y,y')$$ where $p=(x,y)$ and $p'=(x',y')$.
Which one is for $\mathbb{C}^2$? I mean, which one is mostly used for $C^2$? and why?My guess is the first one, and we take $d_{\mathbb{C}}(x,x')=|x-x'|$. But I don't know why, there must be some reason to choose a frequently-used metric.
For example, in most cases we use the first product metrics to define $\mathbb{R}^n$, instead of the other two, because some theorems are built on it. I can think of some advantages of using the Euclidean metric for $\mathbb{R}^n$. So I think, there should be some "bad consequences" if we pick a "bad" metric for $\mathbb{C}^n$. What are these "bad consequences"? Or are there any advantages for choosing a "good" metric?
It is somewhat a soft question. If my question is not clear, please feel free to point it out and I will restate it.
If you just want to prove that, say, bounded closed sets are compact, then there is no difference between these. If you want to discuss more refined structures, then there is.
For example, orthogonality: orthogonal projection onto a subspace, orthogonal complement of a subspace. If you want those, you need the Euclidean norm, not the other ones mentioned. The Euclidean norm is compatible with an inner product, the others are not.
Then there are properties like strict convexity and uniform convexity. $d_{\max}$ and $d_{\rm sum}$ fail those too. These properties were not invented for the fun of it; their lack is a hindrance, e.g., in variational arguments.
A geometrically desirable feature of a metric space $(X, d)$ is: for every two points $a, b\in X$ there exists a unique map $f\colon [0, d(a, b)]\to X$ such that $f(0)=a$, $f(d(a, b)) = b$, and $d(f(t), f(s)) = |t-s|$ for all $t, s$ in the domain of $f$. That is, $f$ is an isometry that maps a line segment into $X$, so that the image of the segment connects $a$ to $b$. Such a curve is called a geodesic, and a space with this property is uniquely geodesic. Of the three metrics mentioned only one is uniquely geodesic: the Euclidean one.