This is my current approach that takes one of the duplicate index values into account when constructing a new column.

import pandas as pd
import numpy as np


def create_shift(df, column, shift_value, method, name):
    """
    Create a new column based on an existing column with a given shift value. 
    The shifted column is indexed based on an existing index with the
    missing values interpolated using the given method.

    :param df:          DataFrame to create the shift in.
    :param column:      The column name.
    :param shift_value: The value to shift the existing column by.
    :param method:      The interpolation method.
    :param name:        The name used for the newly created column.
    """
    if column in df.columns:
        current_index = df.index
        # creating the shifted index with the 2 decimal point precision
        shift_index = [round(i + shift_value, 2) for i in current_index.values]
        shift_data = pd.Series(data=df[column].tolist(), index=shift_index)
        # removing possible duplicates
        shift_data = shift_data[~shift_data.index.duplicated(keep='first')]
        shift_index = shift_data.index
        missing_index = current_index.difference(shift_index)
        combined_index = pd.Index(np.append(shift_index, missing_index)).sort_values()
        combined_data = shift_data.reindex(combined_index)
        combined_data.interpolate(method=method, inplace=True)
        df[name] = combined_data
    else:
        print("[Warning] Cannot create shift {} for missing {} column...".format(name, column))


d1 = {'a': [4.0, 5.5, 5.5, 6.0, 8.5], 'b': [1.0, 2.5, 2.5, 3.0, 5.5]}
df1 = pd.DataFrame(data=d1, index=[0, 1.5, 1.5, 2, 4.5])
create_shift(df1, 'a', 0.5, 'linear', 'c')
print(df1)

I think, this is, what you try to achieve:

#define the shift value
index_shift = 0.5
#find values common to both indices before and after the shift
ind_intersect = df.index.intersection(df.index + index_shift)
#create new column
df["c"] = np.nan
#transfer values from column a to column c
df["c"][ind_intersect] = df["a"][ind_intersect - index_shift]

You can of course fill your new column with other values than numpy NaN.