Absolute nonsense. The only thing I would do differently is that (for SEO purposes at least), the home link should just be "/" with no filename - this is so that search engines don't end up indexing "/" and "/index.php", with the risk that your link juice is split between the two.
All search engines care about is where the link takes them, they really don't care whether it's an absolute or relative link, just so long as it takes them to the right place.
As Max says, dumb rumours like this surround SEO and there are snake oil salesmen galore ... but the truth is, there is no silver bullet.
When you come across a rumour like this, ask yourself "Would it make sense for Google to do this?". Google is in the business of giving people the best websites to match the search query they put in. End of. They are not going to put arbitrary barriers in the way of that, like preferring absolute links. Would giving preference to sites with absolute links give more relevant search results? No. Is Google going to do that? No. Will using absolute linkshelp Google to spider and index your site? No. Is there any SEO benefit from using absolute links? I'll let you answer that, and see if you've spotted the pattern