Allow the analysis and other date filters to provide the option to paste and try parsing the date based on matching common patterns (maybe even allowing user to define options in config file).

For example, the standard SQL datetime2(0) of yyyy-MM-dd HH:mm:ss pasted would convert to the date it parses.

Maybe even consider taking a page from the freaking great design Simple Bank has started applying which is a text based filter applying friendly syntax. I have found this incredibly useful.

They let you type in the search filter

"last 3 months"
"last week"
"date:1/1/2015-1/15/2015"
"date:1/1/15-1/15/15"

and it immediate updates the reports. This is incredibly intuitive after the initial few minutes curve of playing around. Could make the urls even easy to share. :-)

Expanding this:

"1/1/2015 1pm-2pm" VS "1/15/2015 4pm-6pm" for example would let you possibly do a baseline comparison quickly without a lot of effort.

Will to expand and provide some examples if this seems promising.