How can I divide single values of a dataframe by monthly averages?

I think it is generally recommended to use Grouper instead of TimeGrouper. Have a look at this. For example, if your column is called Date, use

grouper = pd.Grouper(key='Date', freq='M')

instead of using TimeGrouper and then continue as @Zelazny7 suggested. If your column is not a datetime index then use

df['Date'] = pd.to_datetime(df['Date'])

First make a grouper:

import pandas as pd

In [1]: grouper = pd.Grouper(freq="1M")

Then make your new column:

In [2]: df['normed'] = df.groupby(grouper).transform(lambda x: x/x.mean())

By passing grouper to the groupby method you group your data into one month chunks. Within each chunk you divide the 15 minute interval datum by the mean for that month.