This is clearly a scam, but what were they hoping to achieve?

[forward]

I received a cold call from a company asking if I still hold shares in X listed company. I do have a very small holding, but I wasn't going to tell them that. They went on to say that they a client who was looking to acquire them and was offering a very good price. I have about £300 worth from years back when the company floated. They are available to anyone at the market price from the Stock Exchange via a broker,what do they hope to gain by offering me more than they are worth?

I don't know who "they" are because I didn't engage with them, but am curious to know what they were trying to do.

I think they were probably just after your bank details - allegedly to pay the purchase money in.

See here - most likely situation here is that they will ask you to pay a 'registration fee' which they pocket and disappear. Alternatively they will persuade you to transfer the shares to them in exchange for a payment that turns out to be worthless (eg forged bank draft).