Niels Bohr : "It is the hallmark of any deep truth that its negation is also a deep truth"

A good example would be the Riemann Hypothesis. Although in this case the opposite is not particularly aesthetically pleasing. P v NP is a great example I think.

The most obvious, rigorous way to define mathematical depth would be the length of the shortest proof from a given set of axioms. However this would include some totally unimportant and arbitrary theorems - one could say that format's last theorem would almost fall into this category - in fact Gauss himself said this.

So the mathematical definition of depth also requires the result to be significant or important. You could define this by saying that many other important results can be derived from it in a reasonably easy way - slightly circular.

Many modern theorems require incredibly long proofs and there seems to be a general idea that depth is related to length of proof and also the number of difficult and significant theories involved in the deduction. I think this is wrong. Most pure mathematical results before 1900 did not require long proofs - virtually all of Euler's results were proved in a few pages. Why then is Euler regarded as a great mathematician. I think its because his mathematics required great imagination and intuition (inspired guess work) to create. Once this is done the proof may not be so difficult. Modern mathematics has perhaps lost some of this - results are often easy to conjecture by analogy with other fields given that abstraction has created a common foundation for many areas - i.e. in algebraic geometry you might try to extend a result from fields to rings. This certainly requires a talent for generalisation but I believe this is easier and more automatic for a highly intelligent individual than Euler's flashes of inspiration.

Apologies for not really answering the question but I hope I have added something to the discussion.