Triangle inequality for complex function

Question

I have to prove triangle inequality for such function: $f(z)=|z|+||\operatorname{Re}(z)|-|\operatorname{Im}(z)||$ so I want to prove: $f(z+w) \le f(z)+f(w)$ Of course it can also be defined for real numbers: $h(x,y)=\sqrt{x^2+y^2}+||x|-|y||$ however I don't see how would it help. I don't know anything about complex analysis that would be other than trigonometric or exponential form of complex number. The problem is that: $g(z) := ||\operatorname{Re}(z)|-\operatorname{Im}(z)||$ doesn't fulfill triangle inequality - $z=2+i $, $w=2-i$ - thus, modulus seems to be important. I've run out of any ideas. Btw. I need it to prove, that $h(x,y)$ is a norm.