Baking – use in place of allspice

The best substitution would be a a simple tube pan.

Besides the unique appearance, bundt and tube pans are used for particularly dense and moist batters. They might be used for a cake that might dry out on the outside before the center cooks through.

If you don't have a tube pan, then there are a couple alternatives. You can use two loaf pans, this should approximate the bundt pan. Another alternative is to use a deep circular cake pan with a ceramic ramekin in the center of it to create the hole. With either method you may need to adjust your cook time a bit depending on your results. (likely 25F lower and baked for a little longer) It also might simply just work without changes.

Cayenne pepper.

I'm actually serious. I haven't tried it in carrot cake but a little capsaicin actually works well with a little sweet to offset it. Chile powder also works well in sweet things.

Cardamom is my wife's favorite and so it goes into many baked goods I make. It would work and be interesting but not spicy.

You can always put in a good extra dose of cinnamon if you want it to punch you in the face.