I think that Corey Chiver’s Mandlebrot Orbits, which plot how the series develop from particular points, are a really interesting way to visualize the Mandlebrot set.

I wrote a version in Javascript to make them easy to explore in your browser.

Enjoy.

Questions or feedback

If you have any questions, feedback or advice that you would like to share on this subject then please don't hesitate to contact me at @joelchippindale.