Here is the answer from someone quite knowledgeable with the regs regarding takeoff alternates. Just thought I would share it:

The flight plan (or flight release) must specify a takeoff alternate if the weather at the takeoff airport is below landing minimums for that airport. Any landing minimums at the takeoff airport that are authorized for the certificate holder may be considered including Category II or III ILS approaches.

The reference is:
§ 121.617 Alternate airport for departure.
(a) IF the WEATHER conditions at the airport of takeoff are BELOW the LANDING MINIMUMS in the CERTIFICATE HOLDER'S OPERATIONS SPECIFICATIONS FOR THAT AIRPORT, no person may dispatch or release an aircraft from that airport unless the dispatch or flight release specifies an alternate airport located within the following distances from the airport of takeoff:
(1) Aircraft having two engines. Not more than one hour from the departure airport at normal cruising speed in still air with one engine inoperative.
(2) Aircraft having three or more engines. Not more than two hours from the departure airport at normal cruising speed in still air with one engine inoperative.
(b) For the purpose of paragraph (a) of this section, the alternate airport weather conditions must meet the requirements of the certificate holder's operations specifications.
(c) No person may dispatch or release an aircraft from an airport unless he lists each required alternate airport in the dispatch or flight release.

So…
If the Certificate Holder’s Operations Specifications authorize Category II or III approaches; and the aircraft and crew are qualified for those approaches; and those approaches are published and operational for that airport; the crew may CONSIDER the fact that they could easily return to the departure airport if there were a problem and therefore they could CONSIDER not having to file a takeoff alternate.

Multitudes of things can happen after takeoff that can degrade the landing capability of an aircraft. It might be PRUDENT to file a takeoff alternate if the departure airport weather is below Category I minimums, but it is NOT required, unless specifically stated by your company procedures.

A takeoff alternate is… by definition… a place to go if you can’t legally land back at your departure airport.

If your Op Specs allow Category II or III approaches and the aircraft/crew/airport are so equipped then there is no LEGAL reason to file a takeoff alternate.

You might also CONSIDER if the aircraft/crew/company is approved to do single engine Category II or III approaches. Many companies are; some are not. But again it is only something to be considered. There are many other things that could go wrong that may necessitate a return to the departure airport.

There may also be further limitations written into your GOM. Or not.

But once again the reg only states:
“IF the WEATHER conditions at the airport of takeoff are BELOW the LANDING MINIMUMS in the CERTIFICATE HOLDER'S OPERATIONS SPECIFICATIONS FOR THAT AIRPORT, no person may dispatch or release an aircraft from that airport unless the dispatch or flight release specifies an alternate…”

So if your Op Specs allow Cat III approaches and the aircraft/crew/airport are qualified and equipped then THAT is the legal criteria to determine if you need a takeoff alternate.

Nothing prevents you from specifying a takeoff alternate just because you want one in case you happen to fry an engine, but that is simply good judgment, not a legal requirement.