It depends why you want status and what your flying patterns are. Southwest has no checked baggage fee, change fee, or cancellation fees for all customers. Also no first class. (But you can opt in to get 4 alcoholic drink vouchers for every 10 one-way trips/5 roundtrips. Or get drink vouchers from the credit card). So while status on other airlines will often get you those benefits, they are already available to all Southwest flyers.
As others have said, Southwest has an open boarding policy, so your choice of seat is based on what's open when you get on the plane, with no pre-reserved seats. So one might want to optimize their boarding position. Generally (excluding pre-boards, family, military boarding, etc), boarding goes in this order:
- Business Select (15 per flight. You can pay for this when you you have to pay for this when you buy a ticket. Often a couple times more than the basic "wanna get away" fare. This fare includes a larger points earning, and a free alcoholic beverage as well as an early boarding position. If all 15 are not sold out by day of departure, you can pay a fee at the gate to be one of the first 15 to board, though
- Customers with Status (A list and A list preferred)
- Customers who purchased "early bird checkin" (it's generally believed that those who bought EBCI earlier will get an earlier boarding position)
- Customers in order of check in, starting at T-24.
Companion pass by itself has no effect on boarding order. It just means someone gets to fly with you for free. (Your boarding number is still determined by the above list). Other benefits of status on A-list are a priority phone line, free same-day standby options, and a small points earning bonus.
Depending on your flying patterns and which airports you'll be flying out of, status may or may not be worthwhile, as one can optimize their boarding position in other ways that may end up being cheaper. IMO if your goal is just to attain status, the credit card is not the best way to get there. (Cheap intra-CA flights might be your friend there).
If you already have status on another airline, you might also be able to use a status match, which makes the answers to this thread a lot easier (discussed here:
New(er) limited promo? A-List/another airlines status)