Do Jehovah's Witnesses have their own version of the Bible?

To simplify everything, yes, the Jehovah's Witnesses do have their own version of the Bible. I would not even really call it a Bible, because it deviates from the original scriptures in many places. An example is where in Colossians 1:16 the word "other" is inserted 4x. The purpose is to make their "bible" agree with the theology that the Watchtower Society teaches, which denies the Trinity. The followers of this religion are taught that Jesus is not God in the flesh and that the Holy Spirit is not a person of the Trinity, but rather an impersonal active force. There are a number of other key beliefs that contradict Christianity. If you compare other translations of the Holy Bible, you will note this, eg., NIV, KJV, NKJ RSV, ESV, etc. The "New World Translation" agrees with none of the others.

Jehovah's Witnesses do have their own Bible (the New World Translation) but they also use other Bible translations.

For example Psalm 83:18 in KJV is at times referred to because it shows Gods name Jehovah.

And you can see how much the New World Translation deviates from KJV, ASV, and Byington in their own online library. For example this link shows Colossians 1:16 with the other translations in the side panel.

I went to the Jehovah's Witnesses Kingdom Hall last weekend and took a photo of a bookshelf in the room from where I followed the program (a room adjacent to the main room from) just because they had a bunch of different Bibles on it (on the left hand side top shelf).

Yes we have our own Bible that we use, but we sometimes do study and compare with other Bibles. I myself have an old pre-1985 King James that still has the name Jehovah in it, as well as 5 other translations (not counting internet access to many more).

The reason we don’t use the King James Version any more (it used to be the Bible the Witnesses used) is because we have an easier to understand Bible with Jehovah’s name restored to it. The old English from the KJV is difficult to understand. The NWT has been translated from the oldest original language manuscripts available into modern English so that the reader can better understand the scriptures. The reason why the NWT was produced is explained in the forward at the beginning of the book.

As for the NIV, that was published and copyrighted in 1973. The Watchtower Society had produced and copyrighted our Bible earlier, in 1961, so naturally by the time the NIV was published we were already using the NWT.

Jehovah’s Witnesses use the standard 66-book Protestant Bible, but usually use their own translation thereof (they do reference other translations from time to time, but generally use The New World Translation). It’s fair to say that the NWT is quite, let’s say, distinctive in places, and has received a fair amount of criticism. The Witnesses do not in any way claim that the translation of the NWT was inspired by God, and are happy to argue their doctrines from other translations if you ask them to. (Indeed, they did so for many years before the release of the NWT, and continue to do so in languages which do not yet have a version of the NWT.)

Here are some of the distinctive features of the New World Translation.

1. In the specific case of John 1:1, which is always brought up in discussions of the translation philosophy of the NWT, it’s probably fair to say that the Greek is a little ambiguous, and the NWT rendering is defensible. They do, of course, provide a footnote and an appendix article on the subject in The New World Translation — with References.
   
   (John 8:58 is another controversial passage with major theological implications, but in this answer I intend to focus more on general themes of the translation, not specific verses.)
2. The terminology is slightly different: what is commonly called the “Old Testament” the Witnesses (and the NWT) call the “Hebrew-Aramaic Scriptures”; what is commonly called the “New Testament” they call the “Christian Greek Scriptures” (the word Christian is intended to prevent any possible confusion with the Septuagint, a Greek translation of the Hebrew Scriptures). This is, arguably, more neutral terminology than the usual. I like it.
3. The Greek text of the New World Translation is the Westcott and Hort master text, not the Textus Receptus used by the King James Version. There’s a certain amount of dispute in Bible translation circles about which text is better. (The NWT is far from the only translation based on the Westcott and Hort text.) Textus Receptus is based on the majority of ancient manuscripts found. Westcott and Hort is based on fewer, but older, manuscripts. The argument in favour of W&H is simply that older manuscripts are probably better. The argument in favour of TR is that the Bible was copied very carefully and the few old texts which happened to survive merely because they were in Egypt, which has a better climate for this kind of thing, were probably inferior copies. Some also claim that if W&H was the better text, God wouldn’t have allowed it to be lost for so many thousands of years. All of this debate (and yes, I have read a book on the subject, firmly in favour of the Textus Receptus and the related Majority Text) is rendered somewhat moot by the fact that most theologians say that few of the differences between the various Greek texts are theologically meaningful. Footnotes reference other texts and ancient translations (including the Vulgate and Syriac translations) in places, but in general the translation is based on W&H.
   
   (The Hebrew text is far more standard. The NWT uses the Masoretic Text, just like almost everyone else. The footnotes of NWT Ref occasionally reference the Septuagint, the Dead Sea Scrolls, and various Syriac translations, but in general they use the Masoretic.)
4. One of the distinctive features of the NWT is the use of the name Jehovah. The Tetragrammaton (four-letter name of God) appears multiple times in the Hebrew Scriptures. Many translations render this as LORD, following the Jewish practice of not pronouncing the Divine Name (though the Jews do write the name in their scriptures). The Jerusalem Bible renders the name as Yahweh, which is a scholarly “best guess” at the original pronunciation. The Witnesses use Jehovah, which is almost certainly not the original pronunciation, but is the traditional rendering in English, found in both religious and secular books for many many years. Certainly including some form of the name is more accurate than bowdlerising it.
5. One of the even more distinctive, and certainly less defensible, features of the NWT is that they also use the name Jehovah in the Greek Scriptures, although it is not found there in any extant manuscript. When the Greek text quotes the Septuagint, they reinsert the name (yes, reinsert, as they maintain that it was there originally). Certainly there do exist editions of the Septuagint which contain the untranslated and untransliterated Tetragrammaton, and others which render the divine name as Pipi, suggesting that they were copied from an earlier version which contained the original Tetragrammaton, the Hebrew letters of which look a little like the Greek letters for Pipi. (I now feel the need for a fantasy novel in which God is called Pipi. It’s a wonderful name.)
   
   However, NWT includes the divine name in other places too. Sometimes support comes from the existence of the Tetragrammaton in Hebrew translations of the Christian Greek Scriptures (some of those Hebrew translations used for support are actually fairly recent, so any support they offer is tenuous at best). The name Jehovah occurs many times in the New World Translation of the Christian Greek Scriptures, each time accompanied by a footnote and an explanation of the rationale in NWT Ref.

In general, the footnotes and appendices in NWT Ref are about the mechanics of translation, not theology. They are about tricky linguistic points and textual variants.