Baking purpleheart?

That would release the oil which would bring the color out, but danish oil should do the same. Try sanding it out too. I wouldn’t put it in the oven unless I was sure it would be okay for the wood and the gases it may release.
For instance purple heart dust is mildly toxic, nausea, malaise can result.
http://www.hobbithouseinc.com/personal/woodpics/_@_toxicity.htm

-- Failure does not stop me, it makes me try harder..... because I'm crazy.

I have heard of using heat to bring out the color however I’ve also heard of this turning the wood different shades of blue as well. However I’ve never seen this verified though I do have distant plans on experimenting with it.

-- --Rev. Russ in NY-- A posse ad esse

Give it up. The purple color is fugitive, and will eventually darken to dark brown regardless of your efforts to stop it.

-- Clint Searl.............We deserve what we tolerate

All exotic woods like padauk, purple heart, will fade to brown, but it’s not that they ever lose their beauty. It’s still very distinctive even after it fades.

-- Failure does not stop me, it makes me try harder..... because I'm crazy.

So, what’s the point of trying to prevent something that’s going to happen anyway? The padauk mantle I made a few years ago is now a nondescript dark dark brown with no character. Bubinga, maple, or walnut with some interesting grain would have been a better choice.

-- Clint Searl.............We deserve what we tolerate

That’s right Clint. But a bit of purple heart in a project sure makes it desirable, while the piece is still new.

To the O.P.- give it a few days, it will return to purple.

-- Dan V. in Indy

I know the purple will fade over time. I was under the impression you could dramatically slow that by using a finish with UV protection. Typically a marine varnish, I think.

My main concern is that I don’t want to start a fire in the oven. But even if that didn’t happen I could see being exposed to 400 degrees warping the wood.

I’m using purpleheart because the person I’m making it for wants that wood and that purple color. Therefore I’d like to bring out the purple color as evenly as I can and then slap finish on it.

I was also under the impression that while purpleheart fades to brown you can also get the purple color to come back, at least temporarily.

Try leaving the purpleheart out in the sun for a while. I’ve heard that can bring the color back.

Leave it in the sun. It will turn purple in a day or so. I do it all the time.
Don’t use anything that has oil to finish it. Minwax Polycrylic works well.