Minima of a complex function. Eh?

Hint:Difference of two sides of a triangle is greater than third side.So |z-(z-2i)|<=|z|+|(z-2i)|

The smallest possible sum of two absolute values is zero. However, it is impossible for both of the above summands to simultaneously equal zero, so the best we can accomplish is for one of them to be zero. This occurs iff $z = 0, 2i$ , or:

$|0| + |0-2i| = |-2i| = 2;$

$|2i| + |2i-2i| = |2i| = 2$

Hence, the minimum value of $|z|+|z-2i|$ is $\boxed{2}.$