In 2022, the category of real or complex Hilbert spaces and bounded linear maps was characterised in purely categorical terms by Chris Heunen and Andre Kornell. The tensor product of Hilbert spaces played a key role in this characterisation. In this talk I will present some work from my Master Thesis which I completed at the end of 2023 with supervision from Steve Lack. I give a new but related characterisation which does not involve the tensor product. As a bonus, we are also able to characterise the category of quaternionic Hilbert spaces.