It is very hard to know without looking at the errors. Do the projects even compile fine?

In any case, latest v1.4 was release on April 2020 and the Elixir release at the time was v1.10. Compatible OTP versions can be found here: Compatibility and deprecations — Elixir v1.20.2

If we want to find first working version then isn’t it easier to just take an elixir value in Phoenix.MixProject.project/0 function located in mix.exs? Following v1.4 branch the minimum Elixir version is 1.4, see:
https://github.com/phoenixframework/phoenix/blob/f8bb00bce84b4d14c7e4614010c1520005f092a9/mix.exs#L10-L10

If we want to be sure which combinations were tested we can simply check travis.yml which contains all supported Erlang and Elixir matrices:
https://github.com/phoenixframework/phoenix/blob/v1.4/.travis.yml#L2-L15

However we need to be careful with them as according to compatibility table:

1. Elixir’s 1.4 and 1.5 supports Erlang version 18, but it wasn’t tested in v1.4 branch (it was tested in all 1.1.x, 1.2.x and 1.3.x versions)
2. Erlang/OTP 20 wasn’t supported before Elixir 1.4.5
3. Erlang/OTP 21 wasn’t supported before Elixir 1.6.6

So the officially supported (tested in Travis CI) versions for Phoenix 1.4.x are:

1. Elixir 1.4.0 - 1.4.4 with Erlang/OTP 19.3 - 19.3.6.13
2. Elixir 1.4.5 with Erlang/OTP 19.3 - 19.3.6.13
3. Elixir 1.4.5 with Erlang/OTP 20.3 - 20.3.8.26
4. Elixir 1.5.0 - 1.5.3 with Erlang/OTP 19.3 - 19.3.6.13
5. Elixir 1.5.0 - 1.5.3 with Erlang/OTP 20.3 - 20.3.8.26
6. Elixir 1.6.0 - 1.6.5 with Erlang/OTP 19.3 - 19.3.6.13
7. Elixir 1.6.0 - 1.6.5 withErlang/OTP 20.3 - 20.3.8.26
8. Elixir 1.6.6 with Erlang/OTP 19.3 - 19.3.6.13
9. Elixir 1.6.6 with Erlang/OTP 20.3 - 21.3.8.24

Note: The phx.new task as well as guides requires Elixir minimum version 1.5.0

Note: Didn’t included -rc.x releaes as they are release candidates, so they aren’t expected to be stable and therefore recommended for production

Note: Some guides mentions Elixir v1.7 (elixircasts) or Elixir v1.9 (hexdocs)

Helpful resources:

1. [phoenixframework/phoenix@8dbb260] Remove old generators and require Elixir v1.4+
2. [phoenixframework/phoenix@49def42] Require Elixir v1.5 for new apps
3. [phoenixframework/phoenix@a56fdfb] Fix the Elixir minimum version on guide to 1.5
4. [hexdocs.pm/phoenix v1.4.18] Installation guide
5. [hexdocs.pm/phoenix v1.4.18] Deploying with Releases
6. [elixircasts.io using Elixir 1.7] #75: Upgrading to Phoenix 1.4

You need to downgrade dependencies in mix.exs as your current mix.exs tries to use the more recent ones. Those dependencies requires newer Elixir versions than phoenix v1.4.

Try to use those commands:

mix archive.install hex phx_new 1.4.18 --force
mix phx.new example