plt.plot needs a list of x-coordinates together with an equally long list of y-coordinates. As you seem to want to use the index of the dataframe for the x-coordinate and each cell contents for the y-coordinates, you need to repeat the x-values as many times as the length of the y-coordinates.
Note that this format doesn't suit a line plot, as connecting subsequent points would create some strange vertical lines. plt.plot accepts a marker as its third parameter, for example '.' to draw a simple dot at each position.
A code example:
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
N = 30
df = pd.DataFrame({f'column{c}':
[np.random.normal(np.random.uniform(10, 100), 1, np.random.randint(3, 11)) for _ in range(N)]
for c in range(1, 6)})
legend_handles = []
colors = plt.cm.Set1.colors
desired_columns = df.columns
for column, color in zip(desired_columns, colors):
for ind, cell in df[column].iteritems():
if len(cell) > 0:
plotted, = plt.plot([ind] * len(cell), cell, '.', color=color)
legend_handles.append(plotted)
plt.legend(legend_handles, desired_columns)
plt.show()

Note that pandas really isn't meant to store complete arrays inside cells. The preferred way is to create a dataframe in "long" form, with each value in a separate row (with the "index" repeated). Most functions of pandas and seaborn don't understand about arrays inside cells.
Here's a way to create a long form which can be called using Seaborn:
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
import seaborn as sns
N = 30
df = pd.DataFrame({f'column{c}':
[np.random.normal(np.random.uniform(10, 100), 1, np.random.randint(3, 11)) for _ in range(N)]
for c in range(1, 6)})
desired_columns = df.columns
df_long_data = []
for column in desired_columns:
for ind, cell in df[column].iteritems():
for val in cell:
dict = {'timestamp': ind, 'column_name': column, 'value': val}
df_long_data.append(dict)
df_long = pd.DataFrame(df_long_data)
sns.scatterplot(x='timestamp', y='value', hue='column_name', data=df_long)
plt.show()
