Ask Mr. Smarty Plants

There is one herbaceous plant that your description brings to mind, Phytolacca americana (American pokeweed).  Although its stems aren't truly woody, they can be substantial and it does grow rapidly and has purple berries.  Here is another photo showing reddish/purplish leaves and you can see more photos here.  It would be nice to know what the shape of the leaves are on your plant and I don't understand whether the leaves near the bottom are purple while the leaves near the top are green or whether the bottom side of each leaf is purple and the top side of the leaf is green.

Since the pokeweed didn't exactly match your description, I did a COMBINATION SEARCH in our Native Plant Database choosing "Florida" under Select State or Province and "Shrub" under Habit (general appearance) to find shrubs native to Florida.  I looked through the 160 results and didn't really see anything that matched your description.   You should try the same search and look at them yourself.  You can also do a similar search using "Tree" for the search term under Habit (general appearance). 

There is a very good chance that your plant is not native.  Florida's climate is very conducive to growing tropical non-natives.  Your best bet for identification would be to take photos and then visit our Plant Identification page to find links to several plant forums that accept photos for identification.