I find zero indexing logical: zero is the first natural number and is thus a fine candidate for being the first ordinal.
In my experience most mathematical series lose nothing in terms of elegance or readability by being indexed from zero instead of using more traditional indexing from one.
I've actually found the opposite, though admittedly it doesn't come up often enough to be a major nuisance.
The reason for this is the set {1...n} has order n, but the set {0...n} has order n+1, so you get lots of off by n+1 inelegancies or errors when order is important. It's better to be explicit at the set level when you need an {0...n-1} set, because usually the order gets passed around to later expressions and not the set element, so there's less algebra.
Zero indexing is great when your index is an offset, as it is for true arrays.
My ideal calendar has 12 30 day months, days 0 to 29.
If it's 10th June and you have an appointment for 0 August, that's 50 days from now.
At the end of the year, a 5 or 6 day 'month' called Holiday. 30th December becomes Christmas Day (observed) and 0 January is still New Years. New Years Eve is either 5 or 6 Holiday.
6 Holiday would be a lot of fun. ^_^