The interpretation of this verse depends entirely on punctuation, which rests wholly on human authority, the Greek manuscripts having no punctuation of any kind till the ninth century, and then it is only a dot (in the middle of the line) separating each word.  See Ap. 94.V. i..3.

The Verb "to say", when followed by hoti, introduces the ipsissima verba of what is said;  and answers to our quotation marks.  So here (in Luke 23:43), in the absence of hoti = "that", there may be a doubt as to the actual words included in the dependent clause.  But the doubt is resolved (1) by the common Hebrew idiom, "I say unto thee this day", which is constantly used for very solemn emphasis (See note on Deut. 4:26);  as well as (2) by the usage observable in other passages where the verb is connected with the Gr. semeron = to-day.

1. With hoti : --
            Mark 14:30 :  "Verily I say unto thee, that (hoti) 'this day ... thou shalt deny me thrice.' "
            Luke 4:21 :  "And He began to say unto them, that (hoti) 'This day is this scripture fulfilled in
                                your ears.' "
            Luke 5:26 :  "Saying (hoti =that), 'We have seen strange things to-day.' "
            Luke 19:9 :  "Jesus said unto him that (hoti), this day is salvation come into this house.' "

For other examples of the verb "to say", followed by hoti, but not connected with semeron (to-day), see Matt. 14:26;  16:18;  21:3;  26:34;  27:47;   Mark 1:40;  6:14, 15, 18, 35;  9:26;  14:25.  Luke 4:24, 41;  15:27;  17:10;  19:7.

2. Without hoti : --

On the other hand, in the absence of hoti (= that), the relation of the word "to-day" must be determined by the context.

Luke 22:34 :  "And He said, 'I tell thee, Peter, in no wise shall a cock crow to-day before thou shalt thrice deny that thou knowest Me.' "  Here the word "to-day" is connected with the verb "crow", because the context requires it.  Compare Heb. 4:7.

It is the same in Luke 23:43 :  "And Jesus said to him, 'Verily I say unto thee to-day [or this day (*1), when, though they were about to die, this man had expressed so great faith in Messiah's coming Kingdom, and therefore in the Lord's resurrection to be its King -- now, under such solemn circumstances] thou shalt be, with Me, in Paradise.' "  For when Messiah shall reign His Kingdom will convert the promised land into a Paradise.  Read Isa. 35, and see Note on Ecc. 2:5.

We must notice also the Article before "Paradise".  It is "THE Paradise", viz. the paradise of which the prophets tell in such glowing language, when the Lord shall come in His Kingdom.  See Ps. 67:4, 6;  72:6, 7, 16, 17.  Isa. 4:2;  30:23, 24;  35:1, 2, 5, 6;  41:18, 20.  Jer. 31:5, 12.  Ezek. 34:25-27;  36:29, 30;  47:8, 9, 12.  Hos. 2:18, 21, 22.  Joel 3:18.  Amos 9:13-15.  Zech. 8:12.

It has no connexion with Babylonian, Jewish and Romish tradition, but is a direct answer to the malefactor's prayer.   His prayer referred to the Lord's coming and His Kingdom;  and, if the Lord's answer was direct, the promise must have referred to that coming and to that Kingdom, and not to anything that was to happen on that day on which the words were being spoken.

It is alleged that the Lord's promise was a reply to the man's thought;  but this is an assumption for which no justification can be found.  Moreover, how can we know what his thought was, except by the words he uttered?

The Lewis Codex of the Syrian N.T. reads in v. 39 :  "save Thyself and us to-day".  So the Lord's word "to-day" may have reference to the revilings of the one, as well as to the request of the other.

(*1)  It is rendered "to-day" eighteen times in the Gospels, Hebrews and James;  but "this day" twenty-three times (five times in Matthew;  once in Mark;  four times in Luke;  nine times in Acts;  once in Romans;  twice in 2Corinthians;  and once in Hebrews).