Do all Christian views need to have a Biblical basis? Stated another way: Is there room for understanding God and/or His will that does not come from the Bible? Perhaps through revelations from God Himself, or secondary sources (Angels, Saints, etc). I would wager the answer to this question depends on the denomination being discussed. But, for example, do the words of the Pope always hold true in Catholicism even if there is no strict biblical basis for them?
A good reference on the reformed, evangelical position on this is from chapter 1 of the Westminster Confession
We speak of the Bible as God's Word because in it he speaks to us (2 Tim 3:16). He speaks to us personally in the Bible through the Holy Spirit (1 Cor 2:12). Because the Bible is considered as God's Word it is his definitive words to us, the only infallible source of knowing him. It is also sufficient for everything we need to know about him. This means that in theory we don't need to look anywhere else to know God's will.
Some points to note:
The Bible won't tell us about absolutely everything that there is to know. It isn't particularly concerned about things like mathematics. The Bible is the final authority on matters of faith. Often it may not even answer your personal questions about God and his will. In this situation, either it means that we need further reflection, or that we're asking the wrong questions which means we may need to realign our concerns with the Bible's.
The Bible won't state everything down explicitly. Often our understanding of God is derived as a consequence of what we read in the Bible. E.g. the Bible doesn't mention the Trinity, but we can certainly see the ideas (e.g. John 17). This has implications on contemporary issues or things which we think the Bible doesn't deal with. We can draw principles and doctrines and apply them to different contexts. But you likely won't find specific answers to decisions in your own context such as whether you should do X or Y.
Not everything in the Bible is easy to understand. Often the difficulty in understanding lies in our own hardness against God, which is why God has given us the Holy Spirit to help us. Often it is because of the difficulty in understanding the actual meaning. The reformers also derived the principle that Scripture should interpret Scripture, which upholds the authority of the Bible while recognising its difficulties. Often the difficulty in understanding is not in individual statements but in the message as a whole and in the emphases. E.g. many parts of the Bible teaches us to love, but it is not the biggest teaching. The biggest teaching is that we need God's salvation because without it we cannot love anyway.
It's commonly recognised that there are at least 3 other sources: reason, tradition, experience. These are part of God's common grace to everyone. They can be helpful, but these should only be considered as secondary sources. They are ultimately flawed because people are flawed.
Reason. We use reason in deducing a lot of things, even reading the Bible. But we know that our capabilities to reason are limited, and people contradict themselves anyway. From a philosophical perspective, we can't know God by sheer reason. Our senses have limited scope, and by nature we deny God anyway (Rom 1:19f). God needs to reveal himself to us and tell us about himself and what he desires. E.g. he needs to tell us that there is only 1 of him. Anything else is just a guessing game. E.g. "I think God is like ..."
Experience. Can we know God from experiencing a vision? We don't particularly need it if the Bible is his revelation to us. A vision doesn't make us more anointed. Neither do we need such a vision or revelation to supplement the Bible since the Bible is sufficient. But if we do get such a vision, it should still be weighed up against Scripture (e.g. 1 Jn 4:1-3). Besides, the Holy Spirit is the important subjective, experience of God. I don't mean speaking in tongues, and it doesn't need to 'feel' particularly spiritual. The fact that we can repent and believe in Jesus is evidence of it already.
Tradition. Can an individual (such as the Pope) have more authority to give a special revelation from God? People may have particular gifts from God, e.g. Eph 4:11. The Apostles carried a unique authority because they were eyewitnesses to Jesus (Acts 1:21). But no one has a special phone line to God. All Christians have the same direct line to God, i.e. the Holy Spirit. Even the Apostles needed that before they could start preaching (Acts 2). Having said that, the teaching of church leaders, books or each other can be helpful. Even non-Christian teachings/writings can be helpful. But that's not to say that they're always right. Neither can they carry the same authority as the Bible, because they are ultimately man's words.
So we can have secondary sources to help us know God, but the ultimate authority is the Bible because of the special authority that the Bible carries as the Word of God. The implication is that everything else is either wrong or inadequate, which means that we do need to come back to the Bible in the end. And because of the difficulty of understanding the Bible sometimes, we need to continually be going back to Scripture. You may even be surprised by what you find.
Clearly not all Christian views need to have a Biblical basis. I'm a Christian, and I have the view that the square of 2 is 4, and that Neil Armstrong walked on the moon, however these view are not explained in the Bible.
I think the closest Christians have to a principle similar to that about which you're asking, is the concept that all Christian views must be consistent with the Bible. And I believe that all Christian traditions can agree with this--to the extent that they believe the Bible can be accurately interpreted.
Catholics hold Tradition in high esteem, but (to my understanding), they believe that Tradition never conflicts with the Bible.
The LDS church has other scriptures they use, but again, these, at least in their view, never conflict with the Bible.
And in my examples, the Bible doesn't say anything about the square of 2 not being 4, nor that Neil Armstrong did not walk on the moon.
I was going to write a nuanced answer, but upon reflection, I think I'll come down firmly on the side of: No.
I realize many will disagree, but I'd like to offer several reasons for my view, as food for thought.
Not all churches that are commonly regarded as Christian believe that all Christian beliefs must have a biblical basis. Most Protestant churches do, but Roman Catholics and Eastern Orthodox give Tradition a strong role. Timothy Ware (aka Metropolitan Kallistos), in his book The Orthodox Church, gives the Bible the principal place within Holy Tradition.
The list of books of the New Testament was not definitively settled until the late fourth century. See, for instance, Development of the New Testament canon on Wikipedia. There were some books that weren't included until very late, while some books were included in early versions of the New Testament that no longer are. For instance, the 5th century Codex Alexandrinus (in the British Museum) contains 1 and 2 Clement, even though the canon was supposedly closed prior to that date. For a long time there wasn't a strict separation between "inspired" and "unspired" books.
Until the invention of the printing press, it wasn't possible for most Christians to possess a Bible. Until Gutenberg, making a copy of a book was a laborious process, potentially a work of months, done entirely by hand. In the past I was told tales of Bibles chained to pulpits in Catholic churches, lest the faithful find out what they really said. In fact, though, Bibles were chained to pulpits because they were so hard to create, and therefore were extremely expensive. Further, the faithful had a chance to hear Scripture at every service; all of the ancient churches had a lectionary, which prescribed readings of the Gospels, Epistles, and in some cases the Old Testament.
But this was a far cry from every man having his own Bible. If Christianity were dependent on that, it could not have come about until the fifteenth century.
Short answer: YES! Why? It's my biblical basis:
2Co 10:5 KJV
Casting down imaginations, and every high thing that exalteth itself against the knowledge of God, and bringing into captivity every thought to the obedience of Christ;
The bible is like a programming language that can compile itself! :)