Why is the total variation of a complex measure defined in this way?

The problem is that $|\cdot|$ is not additive. Consider the complex measure $$ \mu=(a+ib)\delta.$$ Whatever the definition of $|\mu|$ is, I expect that, in this case, it should boil down to $|\mu|=\sqrt{a^2+b^2}$. With your expected definition $|\mu|:=|\mu_1|+|\mu_2|$, however, we would get $|\mu|=|a|+|b|$.

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This example might suggest the definition $|\mu|:= \sqrt{ |\mu_1|^2+|\mu_2|^2}.$ This is not good either. If $\mu=(a+ib)\delta_x + (c+id)\delta_y$, with $x\ne y$, then we expect $$ |\mu|=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}.$$ With the above definition we would instead have $|\mu|=\sqrt{(a+c)^2+(b+d)^2}$.