If you check your ticket, it will say something like 'all passenger must be at the gate and ready to board x minutes before the scheduled time, or they forfeit their flight'.
I have seen times required between x=10 and x=60 minutes, but never less than 10 minutes. In other words, if you miss this limit, it's your own problem, and they don't owe you anything.
For Alaska Air, it is here: https://www.alaskaair.com/content/travel-info/at-the-airport/airport-boarding-times
Boarding:
You must be checked in with a valid boarding pass at the gate no later than 30 minutes before your flight. Being late may cause the cancellation of your reserved seats and/or your entire reservation.
You typically can apply for a refund of the tax and fee part of the ticket price, as they haven't paid those taxes/fees if you are not on the flight, but the remaining part of the ticket might be lost (as are all follow-up legs of the flight), depending on the airlines conditions.
If you ask at the counter right away and nicely, many airlines will get you on the next flight, for a small fee or even for free. But they do not owe you that.
