Col 2:9 is a commonly used proof-text to support traditional Trinitarian formulations of Christology. The text reads as follows:
ὅτι ἐν αὐτῷ κατοικεῖ πᾶν τὸ πλήρωμα τῆς θεότητος σωματικῶς
For in him [Christ] dwells all the fulness of deity bodily (my translation)
Firstly, it should be noted that, in Pauline Christology, Jesus has been exalted after His ascension, and was given a name above all other names:
Wherefore God also hath highly exalted him, and given him a name which is above every name. That at the name of Jesus every knee should bow, of things in heaven and things in earth, and things under the earth. And that every tongue should confess that Jesus Christ is Lord, to the glory of God the Father. (Phil 2:9-11)
This is supportive of LDS "Kingship Monotheism"; in the words of Blake Ostler:
[K]ingship monotheism is maintained because the honor that is given to Christ by bowing the knee to him and confessing that he is Lord ultimately redounds "to the glory of the Father" (Phil 2:11). Just as recognizing the agent or mediator of a benefactor also constitutes recognition and honor given to the benefactor, so bowing the knee to confess Christ ultimately honors the Father. The distinction between the Father and Jesus Christ is maintained clearly because Christ as a mortal is exalted by the power of God the Father. The Father "gives" the divine name that properly belongs to the Father to Christ. Christ becomes the servant who obeys the Father by undergoing death. Further, any confession that Jesus is Lord is ultimately to the honor of the Father. Any notion that Jesus Christ is somehow "included in the unique identity of the One God," the Father, must overlook all of these essential distinctions and names entirely the ways in which such honorific titles as "Lord" and the reception of the name function in a culture of honor and shame. Christ is the recipient of honor; the Father is the one who honors. The Father exalts; Christ is exalted. The Father gives; Christ receives. While the Father and the Son share the same name, glory, exaltation, and honor, Christ is not seen as identical with the one God, the Father. The identities of giver and receiver are clearly differentiated. (Blake T. Ostler, Exploring Mormon Thought, vol. 3: Of God and Gods [Salt Lake City: Greg Kofford Books, 2008], 141-42)
Even after being the recipient of such a name (which implies status, authority, position, and reputation), Christ is still subjected to God:
But I would have you know that the head of every man is Christ, and the head of the woman is the man; and the head of Christ is God. (1 Cor 11:3)
Further evidence that Paul held to a subordinationist Christology, even after the ascension and exaltation of Christ, can be seen in other places in his epistles, most notably his Midrash of Psa 110:1 (109:1, LXX) in 1 Cor 15:22-28.
The Hebrew of Psa 110:1 reads:
נְאֻם יְהוָה לַאדֹנִי שֵׁב לִימִינִי עַד־אָשִׁית אֹיְבֶיךָ הֲדֹם לְרַגְלֶיךָ
Yahweh said to my lord, “Sit at my right hand until I make your enemies your footstool” (my translation)
The LXX (109:1) renders the verse as follows:
εἶπεν ὁ κύριος τῷ κυρίῳ μου κάθου ἐκ δεξιῶν μου ἕως ἂν θῶ τοὺς ἐχθρούς σου ὑποπόδιον τῶν ποδῶν σου
The Lord said to my lord, “Sit at my right hand until I make your enemies your footstool” (my translation)
Here, the first Lord (in the Hebrew, Yahweh) says to a second lord (אדֹנִי adoni in Hebrew, meaning “my lord”) to sit at his right hand. The only meaningful, and exegetically sound interpretation of this verse is that the second lord is sitting at the right-hand of God, making him distinct from "Yahweh," and not that he is numerically identical to the "One God," a la Trinitarianism, though he does indeed serve as God’s vizier, to be sure.
I am aware that some (e.g., James R. White) has tried to argue that the second Lord is Adonai (a substitution for Yahweh), not Adoni, but the LXX, the Targums, and other lines of evidence support the Masoretic vocalisation; for instance, the Targums always interpreted the second lord to be a Davidic King, not "another" Yahweh. For more, see David M Hay, Glory at the Right Hand: Psalm 110 in Early Christianity (SBL: 1973) and Jaco Van Zyl, "Psalm 110:1 and the Status of the Second Lord--Trinitarian Arguments Challenged," in An E-Journal from the Radical Reformation: A Testimony to Biblical Unitarianism, winter/spring 2012, pp. 51-60.
For instance, in The Psalms Targum: An English Translation (ed. Edward M. Cook), the ancient translators/interpreters understood the second lord not to be numerically identical to the One God:
Composed by David, a psalm. The LORD said in his decree to make me lord of all Israel, but he said to me, "Wait still for Saul of the tribe of Benjamin to die, for one reign must not encroach on another; and afterwards I will make your enemies a prop for your feet." ANOTHER TARGUM: The LORD spoke by his decree to give me the dominion in exchange for sitting in study of Torah. "Wait at my right hand until I make your enemies a prop for your feet." ANOTHER TARGUM: The LORD said in his decree to appoint me ruler over Israel, but the LORD said to me, "Wait for Saul of the tribe of Benjamin to pass away from the world; and afterwards you will inherit the kingship, and I will make your enemies a prop for your feet."
In Trinitarian theology, as mentioned previously, there is an allowance (albeit, an ambiguous one) for a distinction between the persons of the Father, Son, and Spirit (e.g. the Father is not the Son). However, there is no allowance for a distinction between “God” and any of the persons. However, the Christology of the New Testament tends to distinguish “God” (θεος) from the Son, not simply the “Father” from the Son, as it does here, differentiating between ο θεος (literally, the God) from Jesus. Indeed, the other instances of the New Testament’s use of Psa 110:1 differentiates, not just the persons of the Father and the Son, but θεος and the Son. For instance, consider 1 Cor 15:22-28 and Heb 10:12-13:
For as in Adam all die, even so in Christ shall all be made alive. But every man in his own order. Christ the firstfruits; afterward they that are Christ's at his coming. Then cometh the end when he shall have delivered up the kingdom to God, even the Father; when he shall have put down all rule and all authority and power. For he must reign, till he hath put all enemies under his feet. The last enemy that shall be destroyed is death. For he hath put all things under his feet. But when he saith all things are put under him, it is manifest that he is expected, which did put all things under him. And when all things shall be subdued unto him, then shall the Son also himself be subject unto him that put all things under him, that God (θεος) may be all in all. (1 Cor 15:22-28)
But this man, after he had offered one sacrifice for sins for ever, sat down on the right hand of God (θεος); From henceforth expecting till his enemies would be made a footstool for his feet. (Heb 10:12-13)
In both these pericopae, Psa 110:1 (LXX, 109:1) is used and expanded upon, and clearly, a distinction is made between, not just the persons of the Father and the Son (which is accepted, albeit, ambiguously, as the definition of "person" is debated within Trinitarian circles, both historically and in modern times, by Trinitarian theology), but God (θεος) and Jesus, a distinction not tolerated by Trinitarianism, as well as showing the Son's subordination, even post-ascension, to God the Father.
With respect to Col 2:9 and its Christological implications, such a promise is also given to believers, paralleling the promise given by Christ in Rev 3:9, 21 that glorified Christians will be the recipients of προσκυνεω (worship) and sit on the Father's throne, just like Christ Himself; such is strengthened by the fact that this is part of a purpose or ινα-clause in the original Greek:
And to know the love of Christ, which passeth knowledge, that ye might be filled with the fulness of God (ἵνα πληρωθῆτε εἰς πᾶν τὸ πλήρωμα τοῦ θεοῦ)
A parallel text in the same epistle is Col 1:19:
Because he [God] was well pleased for all the fullness (πᾶν τὸ πλήρωμα κατοικῆσαι) to dwell in him [Christ]. (Lexham English Bible)
Two aorists are used in this verse (ευδοκησεν [thought/pleased] and κατοικησαι [dwell]). If we go along with the Trinitarian view, then at what point in time was God the Son filled with God's fullness, and was he God before this happened in their view? As we have seen in our discussion of Phil 2:5-11, such is consistent with LDS theology, but at odds with Trinitarian theology. Related to this, in verse 18, we read: "And he is the head of the body, the church is the beginning of the firstborn of the dead, that in all things he might have the preeminence." This is nonsense in light of the hypostatic union which states that Christ was "fully God" while incarnate, but only "veiled" his divinity. However, this is part of a ινα clause in Greek, meaning that Christ became the "firstborn of the dead" in order that he might have the preeminent, or "supreme" (πρωτεύων).
Commenting on the term translated as “Godhead” in the KJV, the following scholarly lexical source comments on this term, and its Christological implications:
Θεοτης – θειοτης
“Divinity,” “Godhead,” in Plutarch, Lucian, Themistios etc., in common in Hermas.
It occurs only once in the NT, Col. 2:9: ἐν αὐτῷ (Christ) κατοικεῖ πᾶν τὸ πλήρωμα τῆς θεότητος σωματικῶς, cf. 1:19f. The Εις θεος [one God] of the OT has attracted to Himself all divine power in the cosmos, and on the early Christian view He has given this fullness of power to Christ as the Bearer of the divine office. (Theological Dictionary of the New Testament, ed. Gerhard Kittel [10 vols: trans. Geoffrey W. Bromiley; Grand Rapids, Mich.: Eerdmans, 1965]; 3:119; emphasis and comment in square brackets for clarification added )
All of these considerations are consistent with LDS Christology (for a fuller discussion, see here) though an examination of the Christology of Colossians and the rest of the Pauline corpus present insurmountable problems to traditional Trinitarian formulations thereof.