def parse_datetime(value):
    """Parses a string and return a datetime.datetime.

    This function supports time zone offsets. When the input contains one,
    the output uses a timezone with a fixed offset from UTC.

    Raises ValueError if the input is well formatted but not a valid datetime.
    Returns None if the input isn't well formatted.
    """
    match = datetime_re.match(value)
    if match:
        kw = match.groupdict()
        if kw['microsecond']:
            kw['microsecond'] = kw['microsecond'].ljust(6, '0')
        tzinfo = kw.pop('tzinfo')
        if tzinfo == 'Z':
            tzinfo = utc
        elif tzinfo is not None:
            offset_mins = int(tzinfo[-2:]) if len(tzinfo) > 3 else 0
            offset = 60 * int(tzinfo[1:3]) + offset_mins
            if tzinfo[0] == '-':
                offset = -offset
            tzinfo = get_fixed_timezone(offset)
        kw = {k: int(v) for k, v in kw.items() if v is not None}
        kw['tzinfo'] = tzinfo
        return datetime.datetime(**kw)