Four test tubes, with unknowns labelled A B C and D contain substances which have approximately the same melting range How could you prove that the test tubes contain four different compounds?

I was thinking that you could purify the samples to ensure that the similar melting ranges for some isn't due to impurities throwing off the melting range. Is this plausible and also are there any other test you would suggest?

Yes, you could do that.

But since you have them in separate test tubes, you can carry out specific tests with them.

Any alkene will decolourise bromine water (aqueous bromine)
Alcohols will be oxidised to carboxylic acids with potassium manganate (you'll see a change in colour of the potassium manganate from purple to colourless if there are alcohols)
Carboxylic acids react with bases like sodium carbonate giving off carbon dioxide.
Alkanes will not react with any of these reagents.

I think that's enough for you. :)

I hope it helped! :)