note: this has a lot in common with Ian Thompson's answer but the approach is different enough to have it be a separate answer. I use the DataFrame format provided in the question and avoid changing the index.
Seaborn and other libraries don't deal as well with datetime axes as you might like them to. Here's how I'd work around it:
Start by adding a column of date ordinals
Seaborn will deal better with these than with dates. This is a handy trick for doing all kind of mathy things with dates and libraries that don't love dates.
from datetime import date
df['date_ordinal'] = pd.to_datetime(df['date']).apply(lambda date: date.toordinal())
Make a plot with the ordinals on the date axis
ax = seaborn.regplot(
data=df,
x='date_ordinal',
y='amount',
)
# Tighten up the axes for prettiness
ax.set_xlim(df['date_ordinal'].min() - 1, df['date_ordinal'].max() + 1)
ax.set_ylim(0, df['amount'].max() + 1)
Replace the ordinal X-axis labels with nice, readable dates
ax.set_xlabel('date')
new_labels = [date.fromordinal(int(item)) for item in ax.get_xticks()]
ax.set_xticklabels(new_labels)
ta-daa!
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