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Which can be an argument against the Lord normally speaking Aramaic, as the this can be seen as recording it in that language because it was an exception to the norm. Which is not the case in Mt. 16:18.

Yet allowing that the normative language was Aramaic, as many evangelicals as Michael Brown does, does not settle the issue (and in any case it also does not translate in the RC perpetuated Petrine papacy, with the church looking to Peter reigning supreme in Rome as its the first of a line of assuredly infallible exalted heads, which is one of the many RC inventions not seen in the NT church)

And adding to the linguistical debate, another researcher finds,

...an Aramaic word-play -- I should say, a possible Aramaic word-play, that nobody really understands -- is foundational to Roman and papal authority.

Both David Garland (“Reading Matthew: A Literary and Theological Commentary on the First Gospel”, New York: Crossroad Publishing, 1995) and Everett Ferguson (“The Church of Christ: A Biblical Ecclesiology for Today”, Grand Rapids/Cambridge: Eerdmans Publishing Co, 1996) point to the 1990 study by C.C. Caragounis, “Peter and the Rock” (Berlin: Walter de Gruyter)

Here’s Garland’s account:

C.C. Caragounis’s study of this passage carefully argues, however, that the rock refers to something other than Peter. The demonstrative pronoun “this” [in the phrase “on this rock”] logically should refer to something other than the speaker or the one spoken to and would be appropriate only if Jesus were speaking about Peter in the third person and not speaking to him. If Jesus were referring to Peter, it would have been clearer to have, “You are Rock, and upon you I will build my church” (Caragounis 89). Petros usually meant a free-standing “stone” that could be picked up; and petrausually was used to mean “rock,” “cliff,” or “bedrock.” But the two terms could reverse their meaning and no clear-cut distinction can be made between the two (Caragounis, 12, 15). If the two words were intended to refer to the same thing, petros could have been used in both places since it could be used to mean both stone and rock. The use of two different terms in the saying, petros and petra, implies that the two were to be distinguished from each other. More
In any case, the linguistical debate is endless and on going, and the answer is to examine what was said in context and how this is understood in the rest of Scripture.
The R.C. exaltation of Peter is foundationally based upon Mt. 16:13-19, wherein there is a play on the word "rock" by the Lord, in which the immovable "Rock" upon which Christ would build His church is the confession that Christ was the Son of God, and thus by implication it is Christ himself. The verse at issue, v.18, cannot be divorced from that which preceded it, in which the identity of Jesus Christ is the main subject. In the next verse (17) that is what Jesus refers to in telling blessed Peter that “flesh and blood hath not revealed it unto thee,” and in v. 18 that truth is what the “this rock” refers to, with a distinction being made between the person of Peter and this rock. This is the only interpretation that is confirmed, as it must be, in the rest of the New Testament. For in contrast to Peter, that the LORD Jesus is the Rock (“petra”) or "stone" (“lithos,” and which denotes a large rock in Mk. 16:4) upon which the church is built is one of the most abundantly confirmed doctrines in the Bible (petra: Rm. 9:33; 1Cor. 10:4; 1Pet. 2:8; cf. Lk. 6:48; 1Cor. 3:11; lithos: Mat. 21:42; Mk.12:10-11; Lk. 20:17-18; Act. 4:11; Rm. 9:33; Eph. 2:20; cf. Dt. 32:4, Is. 28:16) including by Peter himself. (1Pt. 2:4-8) Rome's current catechism attempts to have Peter himself as the rock as well, but also affirms: “On the rock of this faith confessed by St Peter, Christ build his Church,” (pt. 1, sec. 2, cp. 2, para. 424) which understanding some of the ancients concur with.
The R.C. exaltation of Peter is foundationally based upon Mt. 16:13-19, wherein there is a play on the word "rock" by the Lord, in which the immovable "Rock" upon which Christ would build His church is the confession that Christ was the Son of God, and thus by implication it is Christ himself. The verse at issue, v.18, cannot be divorced from that which preceded it, in which the identity of Jesus Christ is the main subject. In the next verse (17) that is what Jesus refers to in telling blessed Peter that “flesh and blood hath not revealed it unto thee,” and in v. 18 that truth is what the “this rock” refers to, with a distinction being made between the person of Peter and this rock. This is the only interpretation that is confirmed, as it must be, in the rest of the New Testament. For in contrast to Peter, that the LORD Jesus is the Rock (“petra”) or "stone" (“lithos,” and which denotes a large rock in Mk. 16:4) upon which the church is built is one of the most abundantly confirmed doctrines in the Bible (petra: Rm. 9:33; 1Cor. 10:4; 1Pet. 2:8; cf. Lk. 6:48; 1Cor. 3:11; lithos: Mat. 21:42; Mk.12:10-11; Lk. 20:17-18; Act. 4:11; Rm. 9:33; Eph. 2:20; cf. Dt. 32:4, Is. 28:16) including by Peter himself. (1Pt. 2:4-8) Rome's current catechism attempts to have Peter himself as the rock as well, but also affirms: “On the rock of this faith confessed by St Peter, Christ build his Church,” (pt. 1, sec. 2, cp. 2, para. 424) which understanding some of the ancients concur with. While men can argue about the significance of the difference between the Greek (the language the Holy Spirit chose to express the New Testament revelation in) words “Petros” (Peter, or stone in Jn. 1:42) and “petra” (rock) in Mt. 16:18, and what the LORD might have said in Aramaic, the phrase “this stone” (“touton lithosis”), used to identify the cornerstone which is the foundation of the church, (Mt. 21:42) is only used of Christ as regarding a person. (Mt. 21:44)
It is by the “rock of this faith” that the church not only exists but it gains its members. (1Cor. 12:13; Eph. 1:13) And it is by the essential faith which Peter expressed that church overcomes: "Who is he that overcometh the world, but he that believeth that Jesus is the Son of God?" (1Jn, 5:5; cf (1Jn. 2:13,14,25) And which Peter himself confirms: "Be sober, be vigilant; because your adversary the devil, as a roaring lion, walketh about, seeking whom he may devour: Whom resist stedfast in the faith, knowing that the same afflictions are accomplished in your brethren that are in the world." (1 Pet 5:8-9)