The obvious dangers here are water-related: drowning or slipping on wet surfaces like the tub or tile floors. Also be conscious of appliances, especially curling irons or hot rollers that can burn exploring fingers or are plugged into sockets near the sink or tub. Look out for potential poisons, such as cleaning supplies, perfumes, cosmetics, vitamins and medications (which can resemble candy), and sharp objects like razors, scissors, tweezers and clippers that can cause cuts or punctures.
Make it safer:
Keep the bathroom door closed, and install a protective doorknob cover so baby can't get in without you. Prevent slips and falls with nonskid, slip-resistant mats. Set your water heater at 120 degrees Fahrenheit or below to avoid scalding (and always test water on the inside of your wrist before bathing). Cover the faucet with a spout guard to protect baby from bumps. Don't use a bath seat -- stay within arm's reach of her, especially when she's in or near water (including the often-overlooked water in the toilet bowl) -- and keep a secure grip when lifting her in and out of the bath. Ensure electrical sockets near the sink and tub have ground-fault circuit interrupters (GFCI), which automatically cut the power if an appliance gets wet -- have them installed if your house was built before 1975, when their use in bathrooms became mandatory. Unplug electrical appliances when not in use, and store them out of harm's way; do the same for countertop temptations, chemicals and the wastebasket. Fit all cabinets with child-resistant latches or locks, and install a lock on the toilet lid too.