Can you grow water plants in a bird bath?

I have a large bird bath with a fountain that the birds use a lot. But the algae is more than I want to deal with. The bath gets shade in the afternoon and full sun in the mornings. Made of concrete that could use painting. I thought of bottom paint but realize it could be toxic to the birds.
Is there a water plant I could grow in the bowl? Too shallow for fish.
Any ideas at all will be appreciated.


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Presuming that it's very shallow, Mimulus.

https://www.google.ie/search?q=mimu...ei=5CxXVYmWKYT9UKL6gKgH&sqi=2&ved=0CAYQ_AUoAQ

 

Are there any that would be just foliage and not flower?

 

Carex oshimensis 'Evergold'

Yellow sedge grows to only 10" tall. Likes its roots in shallow water.

 

I also think it will be too shallow, and you have to really research the plants you plan to plant.

 

I made a small container pond in a child's bath, which isn't very deep and put in a Scirpus Cernuus (known as the 'fibre optic plant') as an oxygenator. Even thought it's very often sold for aquariums, it's hardy to -10ºC and can be either partly or fully sumerged. The birds happily use the 'pond' to bathe and drink, using the plant as a ledge.