What you're looking at here are two incredible examples of a phenomenon known as fano flow, which show non-newtonian fluids appearing to defy the laws of physics...only they aren't defying any laws at all.
What these fluids are defying are the forces of gravity. Both of the fluids in the video have been treated with a water-soluble chemical compound known as polyethylene oxide (PEO). PEO is a polymer, meaning it's made up of a bunch of identical molecules that fit together to form a much larger molecule, sort of like metal links fitting together to form a chain.
Gather enough of these molecular PEO chains together, and the chains themselves will begin fitting together in a process known as cross-linking. In both of the examples featured in the video, the polymer chains in the fluid are bound to one another at strengths exceeding those found between the molecules in liquids most of us are more familiar with, like water.
They're so strong, in fact, that the cross-linked polymers are able to exert a force that counteracts the weight of the fluid itself (at least for a little while), allowing the flow to continue upward, either into the syringe or out of the beaker — gravity be damned.
Oh, non-newtonian fluids... will you ever stop being awesome? Something tells me the answer is no.