I asked myself a question while reading the post: Absolute value of complex numbers $|a+bi|$ .
Let us take Euclidean space $\mathbb R^2$, with dot product noted $\langle .,.\rangle:\mathbb R^2 \times \mathbb R^2\to \mathbb R$ defined by $$\forall x=(\xi,\eta),y= (\xi,\eta)\in \mathbb R^2, \langle x,y\rangle:=\xi\xi'+\eta\eta' $$ According to the Pythagorean theorem learned in school, it is natural to define$$||x||:=\sqrt{\langle x,x\rangle}=\sqrt{\xi^2+\eta^2}$$
Let us then define the symmetric bilinear defined positive form on $\mathbb R$ by $$\langle .,.\rangle:\mathbb R \times \mathbb R\to \mathbb R $$ $$(x,y)\mapsto xy$$ where $xy$ is the product of $x$ and $y$ in $\mathbb R$.
We then ask$$|.|:\mathbb R\to \mathbb R^+$$ $$x\mapsto \sqrt{\langle x,x\rangle}\text{ (definition)}$$Thus we find the famous formulas$$\forall x\in \mathbb R, |x|=\sqrt{x^2}$$ $$\boxed{|x|=d(x,0)} $$where $d$ is the distance on the real line.
From which it is derived that$$\text{Property :} |x|=\begin {cases} x&\text{ if } x\geq 0\\ -x&\text{ if } x< 0\end{cases}$$ In particular, as $|.|$ is an euclidean norm, we obtain directly $$\forall x,y \in \mathbb R, |x+y|\leq |x|+|y|$$ Would there be any downside to abandoning any other definition of $|.|$? On the contrary, wouldn't there be the advantage of avoiding the confusions that some people maintain when writing things such as $$"|a+ib|=\sqrt{(a+ib)^2}"?$$ This is by emphasizing that $|.|$ is a distance$$\forall a,b \in \mathbb R, \text{ writing }z=a+ib=(a,b), \boxed{|z|}=|a+ib|\boxed{=d(z,0)}=d((a,b),0)=a^2+b^2$$
I hope that my questions will not seem totally uninteresting to you.