Official solution
Given that \[\left| {z - {4 \over z}} \right| = 2\]
Now \[\left| z \right| = \left| {z - {4 \over z} + {4 \over { - z}}} \right| \le \left| {z - {4 \over z}} \right| + {4 \over {\left| z \right|}}\]
\[ \Rightarrow \left| z \right| \le 2 + {4 \over {\left| z \right|}}\]
\[ \Rightarrow {\left| z \right|^2} - 2\left| z \right| - 4 \le 0\]
\[ \Rightarrow \left( {\left| z \right| - {{2 + \sqrt {20} } \over 2}} \right)\left( {\left| z \right| - {{2 - \sqrt {20} } \over 2}} \right) \le 0\]
\[\left( {\left| z \right| - \left( {1 + \sqrt 5 } \right)} \right)\left( {\left| z \right| - \left( {1 - \sqrt 5 } \right)} \right) \le 0\]
\[ \Rightarrow \left( { - \sqrt 5 + 1} \right) \le \left| z \right| \le \left( {\sqrt 5 + 1} \right)\]
\[ \Rightarrow {\left| z \right|_{\max }} = \sqrt 5 + 1\]