Introduction:
To determine whether the set M = {[a -b b a] | a, b ∈ R} is isomorphic to the complex numbers under addition, we need to analyze the properties of both sets and check if there exists a bijective function that preserves the addition operation.

Isomorphism:
An isomorphism between two algebraic structures is a bijective function that preserves the operations between the elements. In this case, we are considering the addition operation.

Complex Numbers:
Complex numbers consist of a real part and an imaginary part, typically represented as a + bi, where a and b are real numbers and i is the imaginary unit (√(-1)). The addition operation in complex numbers is defined as (a + bi) + (c + di) = (a + c) + (b + d)i.

Set M:
The set M consists of 2x2 matrices of the form [a -b b a], where a and b are real numbers. In order to determine if M is isomorphic to the complex numbers, we need to find a function that maps the elements of M to complex numbers in a way that preserves addition.

Mapping:
Let's consider a mapping function f: M → C, where C represents the set of complex numbers. We can define f as f([a -b b a]) = (a + bi).

Preserving Addition:
To check if the function f preserves addition, we need to verify whether f([a -b b a] + [c -d d c]) = f([a -b b a]) + f([c -d d c]).

Proof:
Let's consider two matrices [a -b b a] and [c -d d c] from M:

[a -b b a] + [c -d d c] = [a + c -(b + d) (b + d) a]

Applying the mapping function f to both sides:

f([a -b b a] + [c -d d c]) = f([a + c -(b + d) (b + d) a])
f([a -b b a] + [c -d d c]) = (a + c) + (-(b + d))i + (b + d)i + a
f([a -b b a] + [c -d d c]) = (a + c) + (b + d)i + (b + d)i + a
f([a -b b a] + [c -d d c]) = (a + c) + 2(b + d)i + a

Now, let's calculate f([a -b b a]) + f([c -d d c]):

f([a -b b a]) + f([c -d d c]) = (a + bi) + (c + di)
f([a -b b a]) + f([c -d d c]) = (a + c) + (b + d)i

Comparing the results, we can see that f([a -b b a] + [c -d d c]) = f([a -b b a]) + f([c -d d c]).

Conclusion:
Since the mapping