For a chapter title in a video, I want to take a still image of a bamboo jungle and modify it in these ways to get a sort of dreamlike effect:

Simulate running into the jungle in slow motion.
Distort the image so it looks like one of those glass shower doors with multi-angled surfaces.
Have a lag so the distortion looks like it takes a couple of frames to "catch up" with the underlying image.

I don't know if I've explained the effect clearly, but I'll pretend I have. So I'm thinking I should start with a fairly large image, create a series of keyframes that simulate the bounce of running in slow motion and also zoom in slowly to show POV movement into the jungle.

Maybe have a compound clip -two copies of the same clip with the same keyframes, but only one distorted and that one retimed to be a bit slower than the other.

I can SEE this thing (in my mind!) but I have to work out the grunt work of it.

Also, I could use FCP X or Motion 5. Is one preferable for this?

Thanks!

If you want a perspective effect that makes it look like you're actually traveling into the bamboo thicket, you'll have to create layers in Photoshop and then put them into a 3D space in a program like After Effects. I'm not familiar with Motion, but I'm guessing you can do the virtual camera motion with it as well.Then you can have a virtual camera movement that appears more realistic than a simple zoom effect.
Applying the other effects should be easier, I believe NewBlue FX has the type of distortion filters you are looking for.

Motion 5 should be able to do this without a problem. It does have 3d camera placements in the program. I have not tried this though.