A little about my background. I have a undergraduate degree in physics which I obtained exactly 20 years back. I also have a undergraduate degree in EE which I obtained about 17 years back. I have worked as a software engineer all my life and currently also pursuing my MBA in Finance. Needless to say that I am a bit rusty in Physics and math but I have always been interested in Physics and recently started reading Physics again after I read (and enjoyed) the theoretical minimum book by Susskind.

I started reading The Road to Reality and so far like it very much. In introducing the contour integration on a complex plane, he mentions a few things that I do not understand.

1. In Chapter 7, page 124 he says (citing a definite integral of a real function between a and b)

But for a complex plane

Question: Why do we have only one path in a real plane but more than one on a complex plane?

2. He says that he will introduce C-R equations in chapter 12 but he then says that the C-R equations tell us that

Question: I do not understand this statement

Could you help? Please be gentle :)

Is there a book on complex analysis (not too thick) that will help me understand Chapter 7 (or later chapters) in Penrose?

Because to go from the real number a to the real number b integrating in the real line you can only go from a to a +dx, then a+2dx...to b.
In the complex plane, being ... a plane, you can go also "up and down" the real line, to the second dimension of the imaginary numbers, so you can follow even a "complex" (not a casually chosen term ) contour from the point (a,0) to the point (b,0) of the plane.

It's true for an "analythic" function, not for all complex functions: if in the interval (a,b) there is a point in which the function is not defined or goes to infinite, the theorem (see down) is not valid.

See "Cauchy theorem".
P.S.
You, as most of people, even with degrees in physics and mathematics, will be able to read that book just up to a certain point, then ... oblivion