A cap. could do it if you need the current for just a short time. ( and actually at a bit less than 12 v ) A good 12v battery would do nicely. Remember conservation of energy.
However, those capacitors are also very physically large.
Consider using a supercapacitor.
These can be rated up to 20 farads(!!) in a package not much bigger than the 10,000µF, but the disadvantage is they are usually only rated for 2.5V or 5.5V so you will have to connect multiple ones in series.
Due to conservation of energy, you cannot get more energy out than in.
You can however do some calculations.
12V, 600mA, 1 second = 7.2 watt-seconds.
12V, 6A, 0.1 seconds = 7.2 watt-seconds.
So to meet the maximum energy output of 7.2 watt-seconds, at 1.5A, you can only go for 0.5 seconds, approximately...
(EDIT: of course, this doesn't take into account the rating of the power supply. It may well fail/trip from taking 6A, you will need a capacitor to smooth the output.)