Okay for $\mathbb{R}^2$ say, I'm quite happy that $d_1(x,y)=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2}$ and $d_2(x,y)=\max(\{|x_1-y_1|,|x_2-y_2|\})$ that the unit ball is a circle in one and a square in the other and whatnot, and use the definition of equiv norms to show it.
$d(x,y)=0\iff x=y$ and
$d(x,y)=k\iff x\ne y$ for $k>0$
is a norm. If $k\ne 1$ I am not sure what the open ball looks like really, it's going to be interesting to see the topology of this metric.
How is this equiv to any other metric, with the square being eqiv to the circle, as you make the square bigger, the circle grows with the same constant and fits around it (both inside and out). This one doesn't.
This is probably more useful for me to think about because topologically it is different, and also definitions like continuity (right now I'm happy with the metric notion of continuity, rather than open sets in topologies, I'll get there) so I really want to know what differs with this metric from "normal" ones that have some notion of distance. I am struggling to imagine and think about this norm.