Perhaps I am misinterpreting the question, but if you mean: What is the maximum, over all $z_1$
and $z_2$, of the minimum number of segments in a polygonal line connecting $z_1$ to $z_2$,
then it seems there is no bound: A very thin annulus needs an aribitrarily large number of segments
to connect diametrically opposed points.

![annulus][1]


  [1]: https://i.sstatic.net/rL9WV.jpg