What's required for a plane certified to fly to Antarctica?
I'm not aware of any specific 'certification' for aircraft operating to/from Antarctica. The air operations to/from Antarctica varies greatly depending on the location of the airstrip and season.

Actually, there are airports in Antarctica that can handle 'conventional' aircraft (atleast seasonally), like the Union Glacier Blue Ice Runway, which has been certified to handle commercial flights. US Army CRREL Report 93-14, Notes on Antarctic Aviation identifies three broad flight categories:


  
  Long trans-ocean flights to and from well developed Antarctic airports.
  Medium-range flights to and from short substandard Antarctic runways with limited facilities.
  Long trans-ocean flights to and from expedient runways that are rough or soft.
  


According to the type, the aircraft requirement can vary significantly. For example, for the first case, (long range) civilian and military aircraft maybe more than enough. From the same report:


  For flights of the first category, exemplified by Christchurch to McMurdo in springtime, any conventional trans-oceanic transports, civilian or military, should be satisfactory as long as they have adequate useful range, with the ability to reach an alternate or to return to the point of departure, arriving with standard IFR fuel reserves.
  
  For medium-range flights to and from short substandard Antarctic runways with limited facilities, the aircraft can be smaller, but it has to be capable of safe over-ocean journeys, with appropriate range and engine-out performance.
  
  For long trans-ocean flights to and from expedient runways that are rough or soft, the aircraft requires both long range and robust, high-flotation landing gear.


The report also lists the desirable characteristics of aircraft for operations in Antarctica, which should explain why aircraft like C-130 and DHC-6 are used the most:


  
  General configuration: High wing; high engines; tee-tail or high tail; low belly; spacious flat-floor fuselage; large doors (tail ramp,side hatch, swing nose).
  Engines: Multi-engine; gas turbine (free turbine turboprop; turbojet or turbofan).
  Landing gear: Wide stance; short legs; low ACN/LCN; large tires; soft tires; multi-wheel, multi-unit assemblies; large wheel-well fair ings (for big wheels or ski retract).
  Speeds: Reasonable cruise speed; slow approach; low stall and landing speed.
  Robustness: Strong airframe, tolerant to vibrations; reliable engines and fuel system; rough-field landing gear; good hydraulics; instruments, avionics and electrical systems resistant to vibration.
  Maintainability: Easy access to problem areas and parts that need frequent changes; spares inventories based on failure probability, easy troubleshooting; access to additional spares at reasonable speed and cost
  


You can see where this is going. The aircraft used in Antarctica are varied, with anything from C-17 Globemaster IIIs to re-manufactured DC-3s in service, De Havilland Twin Otter being one of the most popular. You can see that most of the aircraft operating from there are high wing aircraft with excellent STOL capability. The report also lists some of the aircraft most suited for operations in Antarctica.



Aircraft with characteristics that are appropriate for Antarctic internal operations; Image from CRREL Report 93-14, Notes on Antarctic Aviation by Malcom Mellor

Most of the aircraft look similar with high wing with wide, short landing gear and multi-engined. Quite a few of these are excellent STOL aircraft too. Obviously, they will be modified with the required navigation equipment and long range fuel tanks (for example, with long range fuel tanks, the range of Basler BT-67 is extended to ~4000 km).

Operating in semi prepared fields require the aircraft to be equipped with skis- like the Lockheed Martin LC-130J. These aircraft have the provision for carrying out JATO operations. A number of smaller aircraft (like the Basler BT-67) are equipped with skis as well.



LC-130 skis, By NSF Contractor - US Antarctica Program Common Share Servers, Public Domain, https://commons.wikimedia.org/w/index.php?curid=5883002

Aircraft operating in Antarctica should also consider the effect of air operations on the fragile and unique environment. The long range of aircraft come in handy as this limits the need for holding fuel reserves at the Antarctic airport, reducing infrastructure requirements and ecological impact.

The aircraft operating to/from Antarctica have, in addition to the usual requirements, items like onboard stairs:


  The A319CJ is designed with ultra-long range tanks, enabling the aircraft to return safely to its original port without the need to land and refuel. .... Skytraders limit the need for extensive ground support infrastructure by equipping its aircraft with specialist remote field functions such as its onboard stairs.




Though quite a few types of aircraft can (and have been) operated in Antarctica, it should be noted that not all of them are equal. A good example would be the conclusion of the inquiry into RNZAF Boeing 757 that landed below published minima at Pegasus Field in 2003:


  ... a whiteout landing. However, in the event of this occurring the likelihood of injury was significantly greater for the Boeing 757, with its long landing gear supporting low-slung engines on a low wing, than it was for the Hercules with its rough-field landing capabilities


This, in addition to the additional navigational aids (TACAN) fitted in the C-13Js. It should be noted that RNZAF still flies 757s to Antarctica.