If I'm running away from chasing gamma photons, I can see them, right? Suppose Bruce Banner goes back in time, convinced that the Hulk is a stupid menace. Let's say he also borrows Quicksilver's speed abilities (maybe the Flash is more appropriate, but based on the recent X-Men movie Quicksilver is plenty fast enough).
Anyway he decides to run away from the burst of gamma rays that turns him into the Hulk (assuming this would actually do the trick). Now obviously, they will still travel at the speed of light. But, due to frequency shifts, shouldn't he eventually look behind and "see" the burst of gamma rays as really just color, starting with violet? If he kept running, would he not see a rainbow cycle of colors? 
Would this mean that the flash could see the whole spectrum in terms of the visible spectrum (that is, he wouldn't see new colors, but new sources of colors), depending of course on his speed?
I guess this is all just a red-shift blue-shift (one shift two shift) problem, so I think it is a valid hypothesis. Am I right, or is Banner doomed to become invincible and mean? 
 A: That's right, running away from a gamma source fast enough would shift them into the visible portion of the spectrum. It goes without saying that he'd have to run quite fast:
$$\frac{\lambda_{\rm obs}}{\lambda_{\rm emit}} = \sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}$$
Picking rough round numbers for gamma radiation at $\lambda=10\;\rm pm$ and visible at $\lambda=100\;\rm nm$:
$$v = 0.99999998c$$
At some point adding more trailing $9$'s just doesn't quite convey the whole idea... $\frac{v}{c} = 1-2\times10^{-8}$ isn't much better though.
As the comments to the question correctly point out, any ambient visible light from sources in front of the runner will be blueshifted into the gamma spectrum (and there are likely a lot more visible photons around than gamma photons), so if the goal is avoiding exposure to gamma radiation it's best to run a bit slower and suffer the UV/X-ray exposure from the redshifted gamma from behind and blueshifted visible from ahead.
On a related topic, I highly recommend this educational video game by a group at MIT. Using some false colour and assuming a slower speed of light they do a great job making some of the unintuitive effects of travelling at relativistic speeds more accessible.
