It may be a short harvesting season, but as you cut a mature bed numerous times between April and June, the plants have to grow and prosper for the rest of the summer to build up the strength of the roots for the next year.
Asparagus makes tall bushy ferns and these would make it hard to plant either side with anything other than shade-loving surface rooted things. Anything that robs the soil of nutrition will affect the asparagus though.
Just one row in a four foot bed sounds very sparing to me. I'd plant two two foot rows, although it's true that the more space a plant has, the larger the spears you will get eventually. (But we like lots of small spears).
Those into companion planting claim that tomatoes can be grown alongside:
"Tomato plants protect asparagus from asparagus beetles. And in turn, asparagus plants have a chemical that has been shown to kill nematodes, a common cause of root ailments among tomatoes." (quote)
Basil is also a good herb to grow alongside and I would have thought this was less likely to rob the soil of too much and is more decorative, especially the 'purple ruffles' types.

I didn't realise there would be any ferns - I imagined a rather bleak and sparse bed after June, and as it'll be the closest thing to my sitting area I want to pretty it up! I was thinking of planting some begonias for a bit of colour. Would that work without upsetting the asparagus?

I also don't like the big thick spears so thanks for that "tip" (sorry, terrible pun). I'm new to veg-planting so please bear with me as I don't always understand when people mention planting distances. Do you mean I can plant 2 asparagus along the 4' width, so they'll be 16" in, with a 16" gap between each other?

Make sure that the soil is free of perennial weeds as asparagus crowns should be left undisturbed and wont like to be dug around inorder to get and long rooted dandelions etc out.

The ferns are the asparagus plant, the bits that photosynthesise and produce all the food for the plant, while the shoots we pick are just the newly emerging stems after winter. The older the plants get, the larger the spears or shoots will get until they've grown to maturity after four or five years.
On planting distances, if you have a four foot wide bed, I'd just plant them in two rows down the bed. Leave about a foot from the edge, and put in a root, allowing about two feet around the roots in all directions. The roots grow enormous over the years and go well down into the soil.
As well as making sure all the weeds have been cleared, as Nickyt08 says, dig in as much composted stuff into the bed as possible as the plants like to be well fed. Each autumn, cut the ferns down after the first frosts, and in late winter pile a six or twelve inch layer of rotted compost on top of the bed. Not only does this give you a material for the new shoots to come through, but it smothers annual weed growth and feeds the bed at the same time.
Good luck, as asparagus freshly picked is a superb crop. Its relatively short season makes you appreciate it even more.

Brilliant - thanks, that's really helpful :)