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How to perform 45 degree phase rotation for a recorded signal?

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I am would like to perform 45 degree phase rotation for a recorded signal.
I know I can achieve 90 degree phase rotation using hilbert transform. but how can I achieve phase rotation of 45 degrees?
I searched a in the internet, I can find examples for basic signals with the function. But, I could not find for recorded signal using matlab.
The figure below shows two signals u and w. the second plot can be obtained be plotting u vs w.
The figure below is the 45 degree phase rotated for u and w and it looks like
I have attached a sample signal similar to the plot. In my case the two signals are z1 and z2.
How can I achieve 45 degree phase rotation?
Many Thanks

Accepted Answer

Addy on 28 Jan 2019
If anyone is wondering, I found a literature.
It is ploarization technique that can be obtained using the following formula.

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