When I used to witness to Mormon missionaries, a question I would often ask them is if I'm already a Christian, why are they asking me to convert again? Most of them didn't have an answer, but one of them cited Luke 22:32, where Jesus says to Peter "...when thou art converted...". Peter, they argued, was already a follower of Christ, but was being told he would convert again. Therefore, he argued, it's reasonable to ask Christians to convert to Mormonism.
It's worth noting that the KJV is the only major version that uses this particular phrasing. The NKJV, ESV, HCSB, and the NIV all say something along the lines of "when you have returned to me". But since Mormons tend to be passionately KJV only, let's stick to the KJV, because ultimately, the other versions only state more clearly to a modern audience what the KJV said to its original audience. To show this, let's read verses 31-34.
“And the Lord said, Simon, Simon, behold, Satan hath desired to have you, that he may sift you as wheat: But I have prayed for thee, that thy faith fail not: and when thou art converted, strengthen thy brethren. And he said unto him, Lord, I am ready to go with thee, both into prison, and to death. And he said, I tell thee, Peter, the cock shall not crow this day, before that thou shalt thrice deny that thou knowest me.”
So Jesus isn't saying there's another gospel on the way (although to concede some ground to the Mormons, Peter hadn't received the real Gospel yet, either). Rather, Jesus is predicting Peter's faithlessness, even telling him that Satan himself is demanding Peter be handed over to him. But Jesus is also predicting that Peter will eventually return to Jesus, and commanding him that when he does, he must "strengthen thy brethren."
This actually gives us further evidence to the context, because we don't just see this prediction, we also see its fulfilment in John 21:15-17: “So when they had dined, Jesus saith to Simon Peter, Simon, son of Jonas, lovest thou me more than these? He saith unto him, Yea, Lord; thou knowest that I love thee. He saith unto him, Feed my lambs. He saith to him again the second time, Simon, son of Jonas, lovest thou me? He saith unto him, Yea, Lord; thou knowest that I love thee. He saith unto him, Feed my sheep. He saith unto him the third time, Simon, son of Jonas, lovest thou me? Peter was grieved because he said unto him the third time, Lovest thou me? And he said unto him, Lord, thou knowest all things; thou knowest that I love thee. Jesus saith unto him, Feed my sheep.” So, by this point, Peter had been converted, and as he had denied Christ three times, Jesus allowed him to confirm his conversion three times, commanding him to "feed my sheep". In other words, "strengthen thy brethren."
So what is it Peter had converted to? We know it can't be Mormonism. There is nothing in the Bible that remotely resembles Mormonism, even in the parts Peter wrote. Rather, Peter had been converted to the Gospel that though we are sinners, Christ died for us, that through faith in Him, we can be given His righteousness, and thus reap His reward. Asking a Christian to convert from that is not Biblical. Rather, it is Mormons who should convert to Christianity.