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 http://cs.smith.edu/%7Eorourke/MathOverflow/Annulus.jpg
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 http://cs.smith.edu/%7Eorourke/MathOverflow/Annulus.jpg
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.
