I don't know of an easy way to do this, but you could make two different paragraph styles, one with each bullet character that you want, and use the Next Paragraph Style setting to have the bullets switch paragraph styles as you type.
This wouldn't dynamically update however, so if you want to insert any bullets in the middle, you would have to reapply the paragraph styles.
Anyone else have a more elegant solution?
You could make a custom font. Use bullet style 1 for A, C, E, G, etc. and bullet style 2 for B, D, F, H, etc. Use numbering instead of bullets and use A, B, C, as the numbering style using the custom font. You might even be able to do this with freeor demo software. This will work for up to 26 bullet points and will update dynamically.
Thanks for your post Alec. I used this process and it worked. However, like you said if I was to delete or add bullet points, it wouldn't dynamically update. I thought there might be a solution using nested styles but under nested styles you can only use multiple character styles (not paragraph styles) so here you would not be able to select a different bullet character.
Scott, I thank you also for your post but I feel there should be a another workaround other than creating a custom font. Forgive my ignorance, but how would I go about creating a custom font? Do I need a special program? Thanks
Although Alec's method doesn't automatically update, there's a shortcut for reapplying the styles in one go rather than having to manually reapply each style in sequence:
1. Select all the paragraphs to be updated
2. In the Paragraph Styles palette right-click on the first style in the sequence and select 'Apply "<name of first style>" then Next Style'.
Scott's method is really quite good, and Jongware is developing a tool to make custom fonts directly in ID, which he just publicly announced today: http://forums.adobe.com/message/4518080
Unfortunately the the current free demo is limited to a single glyph so it won't solve your problem, yet, but keep an eye open for the full version, hopefully coming soon. I have used the demo to create custom bullets and glyphs from client logos, and it is fantastic.