For a prime $p\equiv 1\pmod{4}$, we can write $p=a^2+b^2=N(a+bi)$. Therefore $$ a+bi=p^{1/2}e^{i\varphi} $$ where $\varphi\in [0,2\pi]$. I know that Hecke proved that $\varphi$ is equidistributed. I am looking for a reference for this nice result. I would be thankful if one can give me a reference.
One reference is Theorem 6 of Chapter XV (Density of Primes and Tauberian Theorem) in
This is probably more general than Hecke's result, but the case of "equidistribution of ideals and primes in sectors" of the Gaussian numbers is singled out as Example 2 on page 318. [No, I didn't know this off the top of my head; my student David Jao needed this result in the case of a real quadratic field for his thesis in 2003, and I looked in the bibliography to find that he used the Lang reference $$ or more accurately its second edition (1994) by Springer.] 


A very downtoearth treatment of this result of Hecke is in Chapter 5 of the nice book Geometric and Analytic Number Theory, by Hlawka, Schoißengeier, and Taschner. By downtoearth, I mean that they deal directly with this specific case of Hecke's result, and that they prove it using very little  the method is a modification of the KorevaarNewmanZagier approach to the prime number theorem, and so doesn't need any quantitative zerofree region (just a statement that there are no zeros of the appropriate objects on the line $\Re(s)=1$). 


If you need the original reference, this is proved in Hecke's articles (here and here) where he introduces the famous $L$functions associated to Grössencharakteren. E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. Math. Z. 6 (1920), no. 12, 1151 ; Math. Z. 1 (1918), no. 4, 357376. Here is an extract from the Zentralblatt review of Hecke's articles :


