The function shifts alphabetical characters by one position, increments digits by 1, and swaps characters at even indices with characters at the next odd indices.

[CODE]:
```python
def approximate_function(input_str):
    shifted_str = ""
    for ch in input_str:
        if ch.isalpha():
            if ch.isupper(): # upper case
                shifted_ch = chr(((ord(ch) - ord('A') - 1) % 26) + ord('A'))
            else: # lower case
                shifted_ch = chr(((ord(ch) - ord('a') + 1) % 26) + ord('a'))
        elif ch.isdigit():
            shifted_ch = str((int(ch) + 1) % 10)
        else:
            shifted_ch = ch
        shifted_str += shifted_ch

    result = "".join([shifted_str[i^1] for i in range(len(shifted_str))])
    return result
```