The bathroom is potentially a very dangerous room for a baby, but if you take a few steps, you can make it far safer.
First of all, remove all sharp implements from the room. Razors, scissors and the like can either be stored in your bedroom or secured in a locked cupboard where your child cannot reach them. The same applies for medication and cosmetics and even for mouthwash, which is high in alcohol.
Other than that, safety will largely revolve around bathing your baby. If you set your hot water heater no higher than 48 degrees Celsius, this will minimise the chance that your baby could be scalded by hot water, but even so, it is worth investing in a bath thermometer. Floating bath thermometers that double as rubber ducks or similar can be found, which mean they can be left in the bath for fun as well. It is also worth running cold water first when running a bath as this also reduces the risk of scalding.
Non-slip bath mats are very useful as they prevent your child from sliding and a bath ring or bath seat serves a similar purpose, as does a bath support. These products will allow your baby to lie unsupported in the water, meaning you have both hands at your disposal. However, it is important to note that a baby should never be left unsupervised in the bath. Such products are merely aids.