I veganised a recipe handed down from my mum, and it's great. It's a lovely moist cake, keeps for a good while, and easy to make from mostly store-cupboard ingredients.
The most exotic thing involved is the ingredient that I used to replace egg: chia seed. You can also try flax seed. This other website has a nice guide on how to make "chia egg" or "flax egg" - there is something in these seeds that makes a glutinous substance that can bind a cake together.
The end result is great. Compared against the non-vegan version there's a different texture - the edge of the cake is chewy/crispy in a nice way, I find.
First prepare the chia egg: grid the chia seeds in a pestle and mortar, mix the water in, and leave it to stand for a few minutes while you do the other things.
Line an 18cm (7 inch) round cake tin with greased greaseproof paper, and preheat the oven to 180ºC (350ºF, gas mark 4).
Put the sugar into a mixing bowl and gradually whisk in the oil, then whisk the eggs in one at a time (easier to get them mixed in smoothly that way). Add the flour, cinnamon and baking powder and stir the mixture well, beating out lumps to make the mixture as smooth as you can. Add the carrots, nuts, and dates, and mix.
Pour the mixture into the prepared tin and bake for about 1 hour 10 minutes, until the cake is risen and firm to the touch.
Remove from the oven, leave to stand in the tin for 3 minutes, then turn out onto a wire tray, peel off the paper and leave to cool.