I have this plant in my garden and it is dying. I would like to buy a new one but I have no idea what it's name is. Please help me find the name.
They are not massively long-lived. A few years usually.
Ceanothus can have a natural life cycle of 10-15 years, depending on the species and\or growing conditions. The article at the link below has some great information and tips re: care, pruning, etc.
Here is a link that might be useful: Ceanothus
In ideal conditions yes, they can grow for that length of time. But 10 years is not that long imo. They are prone to frost damage and do not respond well if you try to cut into older growth to rejuvenate. As the OP says his is not healthy.
We don't know where the OP is situated but I took a wild guess from the other plants visible and the glimpses through the foliage that this plant is in Europe and possibly in the UK. They will grow quite well here but dwindle after some years just as the pictured plant has.