Coleman Rechargeable Air Pump
Coleman makes various models that look similar, but this is the rechargeable 12V + 240V air pump model.
This popular cordless air pump has been tried and tested for years and is still a common purchase amongst Aussie campers. It may be a little higher in price than some of the other air pumps, but you'll rarely find something it can't do.
It has an inbuilt battery which you can recharge through your car's 12V DC cigarette socket, or through a normal 240V wall socket (both are regulated to 7.5V / 0.2A) using the adapters provided. While I haven't done an extensive test to see how long the battery lasts on a single charge, I would expect it to run for around 20 - 35 minutes on single charge which is plenty. You're unfortunately not able to pump up your mattress or kayak while charging the unit and being a lead battery it's recommended you fully charge the unit prior to your first use.
While you won't be pumping up car tyres with it because of its high volume, low-pressure design but it does come with two separate nozzles and I haven't had any trouble fitting them on different air mattresses, inflatable pool toys etc.
The Coleman QuickPump Rechargeable is also capable of deflating by attaching the nozzle to the inlet valve instead.
This is a really solid choice for a rechargeable and versatile air pump, however, competitors have overtaken the Coleman QuickPump in design and portability. If your needs don't stretch too far outside of inflating and deflating products and you've got the foresight to charge this before your camping trips, then I would say you can't go wrong with this air pump.
Pros
Design - Sturdy design that has stood the test of time.
Versatile - Comes with 2 different nozzles to fit most inflatable valves.
Cons
Noise - It can be a bit loud so be mindful of the time and proximity when using.
Dated - Innovation means there are other models on the market which offer much better value for money