Not too sure about that. The frame sticker looks to be in pretty good shape. Those take a beating if submerged. But "ya neva know"...

 

Looks ok to me.

Have the mechanic poke everywhere w/a screwdriver if it goes through:

 

This last picture you posted is more revealing I do see some flakes in the shackle area. Poking those area's that look like this with a screw driver is a good idea, I would also put a wire wheel on a drill and hit those area to see whats underneath the rust.

Honestly this is one of those subjective things, You can ask a number of people is this bad and you will get a number of answers.

Get under there yourself and check it, Bottom line considering you are buying 15 year old vehicle this will be the least work you will have to do and right now the most important BEFORE you buy it! Good luck!

 

My 09 came from South West Arkansas and has virtually zero rust, the frame is still OEM black. Those pics have bubbling rust and discoloration on the chassis and the frame paint is just gone. It does not look deep enough to be structural, but certainly needs big time inspection.

 

I would walk away and not look back.

 

I would not buy it. My 07 H3 that my parents bought me in 06 doesn't have a lick of rust to this day. It's a Georgia vehicle. Looks like it just got out of the Hummer Dealership (RIP).
My H3T is a upstate NY truck that I purchased used...It has rust. It makes every repair 10 times harder. Brakes, shocks, cv axels, hell even replacing the taillight bulbs are problematic. I have to trade it in for a southern h3t when I can. Sure the T gets compliments daily but they don't know what I know...anyway, anyone want to buy my T?

 

I would agree with H2-SUT. Mostly surface rust. Good Alphas are getting more and more difficult to find. Wire brush any loose rust and use a rust encapsulator on the frame. As a car restoration hobbyist I/we have been using POR-15 for many years. It's the best option to brush on the frame. 2 coats are best. For the body panels, get to the rust under the paint, stabilize it and then you can paint over it if you like. It will take some time but then it will be protected for many years. Besides, finding a rust-free frame on a 16 /15 year old Alpha would be difficult and would likely be from an area far away from the east coast, more likely the southwest. Youd spend more time and money just tracking it down.

 

Rust oleum has many great rust abating products to with preventive maintenance can avert many problems.

Little wire brush and spray on can kill most superficial oxidation issues.

Test for level of rust problem and proceed.