I am lucky enough to spend a bit of time each year in South Island.
It is not uncommon to see big eels when trout fishing in rivers of all sizes, even quite small streams. If one comes close by while you are wading it can give you a fright. They are huge, and not timid. A friend was cleaning a trout of about three pounds and a big eel gently took it away from him.
I've never seen anybody fish for them. I'm not certain, but I think they might be protected.
The biggest of all are said to be in the big hydo lakes, under the turbines. Apparently they just sit there chomping up chopped up trout and salmon. Reputed to get to 100ish lbs there.
However, someone said earlier in this thread "I can catch brownies anywhere". Not like NZ browns you can't! This is sight fishing for big, wild, difficult, fish, and angling doesn't get much better.