If I purchase a company with depreciated assets do I depreciate them further from the current book value or do I depreciate them from their new value?

For historical cost purposes, assets are recorded on the balance sheet at their original cost; this is called the book value. Depreciation is not taken out of these assets directly. It is instead recorded in a contra asset account: an asset account with a normal credit balance, typically called "accumulated depreciation". Balancing an asset account with its corresponding accumulated depreciation account will result in the net book value. The net book value will never fall below the salvage value, meaning that once an asset is fully depreciated, no further expenses will be taken during its life. Companies have no obligation to dispose of depreciated assets, of course, and many depreciated assets continue to generate income.