Washing machine drain hose very high

I would very much agree with your assessment of the situation. Once the washer has pumped all the water out of the tub, it's done pumping. Water left in that hose will run back down and back into the tub, leaving your clothes wet again.

I see three possible solutions:

Good

A second run of the spin-cycle may get a reasonable amount of water out of the clothes and take less time in the dryer.

Better

Find a floor drain. Run your washer hose to a 5-gallon bucket and affix the hose so it drains into the bucket. Cut a hole (3-4" should do it) in the bottom of the bucket near one edge to allow water out. Place the hole in the bucket over the floor drain. Weigh the bucket down with a cinder block placed across the top. Water will drain into the bucket then run directly into the floor drain. The bucket will provide a sump/cache to hold water flowing from the washer in case it won't drain quite as quickly as the washer pumps it out. The bucket will also provide splash containment to help prevent water from spreading too far across the floor. Potentially significant drawback: If the water comes out too fast and/or the bucket gets moved, you can end up with water spread all across the floor. This appears to be a rough, unfinished basement with a concrete floor. Some water on the floor isn't likely to damage the structure, you just have to ensure you're not storing anything on the floor that should be kept dry.

Even Better

Put a check valve on the back of the washer between the washer's outlet and the existing hose. This will allow the washer's pump to force water out of the tub, up hill and into the drain. When the pump kicks off, the valve will seal the drain pipe preventing the water from flowing down hill and back into the tub.

Best

Get the washer to drain below where the outlet from the washer is, this will allow the hose to drain into the sewer instead of back into the washer. It appears that one of these might be a clean out or simply be a capped off bit of drain plumbing with a trap:

If that's the case, you may be able to unscrew the lid and have the washer drain into one of them. NOTE: You MUST have a trap after the opened pipe. If you don't you'll get sewer gas backing up into the basement. Aside from being smelly, it's potentially explosive! If you're not sure, I'd recommend that you ask the landlord about the pipes.

Two things I see wrong with your setup.

First, the white standpipe that the washer drains into needs to be a minimum of 30" (I think) tall.

Second, I believe there's a maximum height above the floor (bottom of the washer) requirement for the top of the standpipe, that I think is 55". Your setup (well, your landlord's setup) violates both those requirements.

I don't see an easy way around this. You could put the washer up on a stand, but that would make access almost impossible.

Best solution is a sewage pump. You drain the washer into a holding tank that's part of the pump. Then the pump pumps the water up to your existing standpipe.

Would be good to see the specific model. But a check valve may be the simplest solution.

From a random Maytag manual:

For standpipe installation above 48"(121.2 cm), install a check valve to prevent water from flowing back into washer. Available from local hardware store.

It's easy enough to see if the pump goes high enough.

and of course prevent backflow (syphoning).

Other consideration is the pump overheating. As the height increases the pump will do more work, and probably get hotter.

Good machines will have a thermal cutoff on the pump, turning it off when it gets too hot, until it cools down. The pump is usually quite cheap.

The cutout is visible (in my experience) on the pump electrics.

In this case, just make sure if the pump gets hot, nothing catastrophic happens - e.g smoke filling the house, lint catching fire.

I have replaced a few pumps, and found it easy to do. So if you can find a better - one go for it.