Called the number on the error page and waited on hold 10 minutes. Hung up and called 1-800-ALASKA-AIR. Was transferred immediately to the Gold Desk (of course) and asked the agent to connect me to the website support group.
She told me to wait while she pulled up the account and then told me, "Wow! It looks like our systems are down. I just spoke with the previous customer and they were just working!"
I explained to her that others were able to log in to their accounts and I could even log into a family member's account with no issues. She then promptly transferred me to Customer Care, who answered in about 30 seconds (instead of the 10 minutes I spent holding earlier).
Same thing transpired. She read my Mileage Plan account number back to me and then said, "It appears our entire system is down, which is why you probably can't log in." I explained to her that no, it was just me
twice and actually tell her I was logging in at that moment to another person's account with no issues before she believed me.
She said she would have to report it and was there anything else she could help me with. I said, "Well, is there any way I can get an expected time of resolution?" She then placed me on hold for about five minutes before coming back and saying that she would have to file a trouble ticket and to try it again in an hour. I asked for a case number in case it still wasn't working later and she said she would make notes in my account that customer care could read.
A bit of a brusque attitude from a supposed customer service professional, but hey, as long as things are fixed, I'm OK.
Interestingly, whichever error message those two agents were seeing when they attempted to log in with my account was telling them (like it did to me) that the whole system was down. Odd...