The surname Purser is an old Saxon name. In those days people did not have fixed surnames that passed on from generation to generation. A baby would be given a single name, chosen from a fairly limited stock of names. This would result in quite a few children of the same name in the same village. If as a forinstance, the local smith had a baby boy  who was named John, to avoid confusion he would be known as John Son of Peter the smith.  His father was known as Peter the smith because smithing was his trade.

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When John son of Peter the smith got old enough to enter a trade himself,  lets say as a carpenter, he would come to be known as John the carpenter, if more clarification was needed, he would be John the carpenter, Peter the smith's lad. or some such thing. For  a married woman it would end up with Er inside, wife of Peter the carpenter etc, etc. On a good day her name might be substituted for Er Inside.

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There was no need for a fixed surname, you were known by what you did, it seems a bit of a mouthful to us but appears to have worked at the time. Mind you, I have not yet come across anyone known as Peter, son of John the lazy bugger. I don't know what they did in that circumstance.

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It was only in later medieval times that more specific family names were needed. There were two main things that drove the change.

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1.  As the population expanded there were just too many people of the same name and in the same trade, especially in the towns.

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2.  As taxation became more sophisticated than just giving your lord a share of your produce or your labour, it became necessary to keep proper written records.

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As surnames became more widespread most people would stick to a name relating to their occupation. So Peter the blacksmith became Peter Smith. Peter the fisherman became Peter Fisher and Peter the lazy bugger became Peter Something suitable.