Pick elements of largest absolute value

(Edited with a slight tweak for a tiny bit more speed)

For the non-duplicate version I am finding this quite fast:

f = If[+## > 0, ##] & @@ {Max[#], Min[#]} &

f /@ list
(* {-5, 6, 4, -9} *)

I'd use Ordering, as Nasser did, but with the default sort as it will be much faster:

# ~Extract~ Ordering[Abs@#, -1] & /@ list

{-5, 6, 4, -9}

If duplicates are required we can substitute Position but Pinguin Dirk's method is faster.

list = {{1, -3, -5}, {2, 1, 6}, {0, 2, 4}, {-9, 2, 6}, {1, -2, 2}}

# ~Extract~ (Position[#, Max@#] &@Abs@#) & /@ list

{{-5}, {6}, {4}, {-9}, {-2, 2}}

---

Here is a concise use of Pick that appears to be competitive for the with-duplicates case:

f[a_] := Pick[a, #, Max@#] & @ Abs @ a

f /@ list

{{-5}, {6}, {4}, {-9}, {-2, 2}}

Timings in version 7 (Pick should be faster in v8 and after):

SetAttributes[timeAvg, HoldFirst]
timeAvg[func_] := Do[If[# > 0.3, Return[#/5^i]] & @@ Timing@Do[func, {5^i}], {i, 0, 15}]

list = RandomInteger[{-30, 30}, {50000, 20}];

# ~Extract~ (Position[#, Max@#] &@Abs@#) & /@ list  // timeAvg
Pick[list, UnitStep[# - Max[#]] & /@ (Abs@list), 1] // timeAvg
f /@ list                                           // timeAvg

0.2496

0.2184

0.1996

Nasser's timings in version 9 are: 0.328, 0.112, 0.203 showing that Pinguin Dirk's code is by far the fastest in recent versions.

And timings for the non-duplicate methods:

list = RandomReal[{-9, 9}, {50000, 20}];

# ~Extract~ Ordering[Abs@#, -1] & /@ list                                  // timeAvg
(SortBy[#, Abs] & /@ list)[[All , -1]]                                     // timeAvg
If[+## > 0, ##] & @@ {Max[#], -Max[-#]} & /@ list                          // timeAvg
Pick[Flatten@#, Flatten[Ordering@Ordering@Abs[#] & /@ #], 3] &@list        // timeAvg
MapThread[Extract, {list, Ordering[#, -1, (Abs@#1 < Abs@#2) &] & /@ list}] // timeAvg
Max /@ Pick[list, UnitStep[# - Max[#]] & /@ (Abs@list), 1]                 // timeAvg

0.078

0.1092

0.128

0.2308

0.842

0.2652

With the clarification, I think I can reuse my solution I initially provided by just mapping Max over it again:

Old:

Pick[list, UnitStep[# - Max[#]] & /@ (Abs@list), 1]

New:

Max /@ Pick[list, UnitStep[# - Max[#]] & /@ (Abs@list), 1]

(note: the old solution returned any value that has max abs value, i.e. {1,2,-2} returned 2,-2).

Based on this new information, this approach is definitely not the best out there, see @Mr.Wizard's answer.