From Osher Doctorow

Sir Roger Penrose is and has usually been a Nonconformist (one has only
to look at his championing of Consciousness, black holes, and twistor
theory for this), but his most remarkable achievement is arguably his
persistence in following through with complex analysis (including
higher dimensional complex spaces) in his twistor program.

I have discussed relationships between Probable Influence/Causation
(PI) and complex analysis in various of my sci.physics threads, but Sir
Roger's recent semi-popular book with many equations (surprising for
books supposed to appeal to the general public), The Road to Reality,
Knopf: N.Y. 2005, with 1099 pages, reveals a surprising humility and an
admission that twistor theory hasn't accomplished most of his key goals
although he is convinced that it is probably correct. But it is also
exceptionally valuable for focusing on the key role of complex analysis
including in higher dimensions in twistors.

What emerges from this volume and from my recent threads on PI seems to
be a startling conclusion or near-conclusion, namely that complex
analysis is a type of code or cryptographics of the Universe which,
although it may be less accurate and less comprehensive than Probable
Influence/Causation (PI), often goes in a similar direction to PI.

A second conclusion is arguably that both PI and complex analysis work
largely through an unexpected characteristic, namely that they are
fundamental to phase changes or phase differences (phase in the sense
of liquid, solid, gas, superfluid, superconductor, plasma, black hole,
liquid crystal, etc.).

In many ways, it helps to look at quaternions for understanding this,
and in fact quaternions and possibly octonions share some of the
characteristics referred to in the last two paragraphs.

Let's look at a complex variable z:

1) z = y + xi, x and y real

The algebra or field of complex numbers doesn't emphasize this overtly,
but if one looks very carefully at (1), a curious characteristic
emerges: namely, that x and y are kept separate by the expression y +
xi. Moreover, they are kept separate in large degree or amount by the
rules of the algebra or field as in addition and multiplication, but
with some curious and yet specifiable interactions.

What am I getting at here? Well, two phases in the above sense differ
by being SEPARATE. The habit of studying Proximity via PI sensitizes
one to this, arguably more than the habit of studying geometry using
metrics.

Now look at this:

2) z1z2 = (y1 + ix1)(y2 + ix2) = (y1y2 - x1x2) + i(x1y2 + x2y1)

where z1 = y1 + ix1, etc. Compare this with:

3) P(A-->B) = (x --> y) = 1 + y - x

and therefore:

4) (x1-->y1)(x2-->y2) = (1 + y1 - x1)(1 + y2 - x2) = (y1y2 + x1x2) -
(x1y2 + y2x1) + U

where U is given by:

5) U = 1 + y1 + y2 - x1 - x2

Notice that except for U, the minus (negative) sign in (4) corresponds
to the i in (2), while the terms with + signs when parentheses are
dropped (opened) correspond to the non-i terms in (2).

From Osher Doctorow