Buying a flat for extra space

Hi

Is this mad?

Been trying to buy a house for a while, proving impossible to get a much bigger place in the right area that we can afford. (London so school catchments very small.)

As a leftfield plan considering staying in our house (which we love, location esp, just a bit small) and buying a flat nearby for extra space (spare room for visiting relatives; hobbies like music and art; etc).

Having numbers it seems to cost no more than upgrading house size; saves on stamp duty; and gives us an asset we could sell one day.

What are we missing? Is it insane? Any hidden legal / costs we should know about? (Realise things like council tax will be higher.)